3. Functions

Intro to Functions & Their Graphs

3. Functions

Intro to Functions & Their Graphs

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Relations and Functions

Verifying if Equations are Functions

Relations & Functions Example 1

Finding the Domain and Range of a Graph

Finding the Domain of an Equation

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Practice this topic

Multiple Choice

State the inputs and outputs of the following relation. Is it a function? { $\left(-3,5\right),\left(0,2\right),\left(3,5\right)$ }

State the inputs and outputs of the following relation. Is it a function? { $\left(2,5\right),\left(0,2\right),\left(2,9\right)$ }

Is the equation $y=-2x+10$ a function? If so, rewrite it in function notation and evaluate at $f\left(3\right)$ .

Is the equation $y^2+2x=10$ a function? If so, rewrite it in function notation and evaluate at $f\left(-1\right)$ .

Find the domain and range of the following graph (write your answer using interval notation).

Find the domain of $f\left(x\right)=\sqrt{x+4}$ . Express your answer using interval notation.

Find the domain of $f\left(x\right)=\frac{1}{x^2-5x+6}$ . Express your answer using interval notation.

In Exercises 1–30, find the domain of each function. f(x)=3(x-4)

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Textbook Question

Without using paper and pencil, evaluate each expression given the following functions. ƒ(x)=x+1 and g(x)=x^2 (ƒ-g)(2)

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Textbook Question

Without using paper and pencil, evaluate each expression given the following functions. ƒ(x)=x+1 and g(x)=x^2 (ƒ∘g)(2)

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In Exercises 1–30, find the domain of each function. g(x) = 3/(x^2-2x-15)

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In Exercises 1–30, find the domain of each function. f(x) = 1/(x+7) + 3/(x-9)

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Textbook Question

Let ƒ(x)=x^2+3 and g(x)=-2x+6. Find each of the following. See Example 1. (ƒ+g)(-5)

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Let ƒ(x)=x^2+3 and g(x)=-2x+6. Find each of the following. See Example 1. (ƒ-g)(4)

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Textbook Question

In Exercises 1–30, find the domain of each function. f(x) = √(x - 3)

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Let ƒ(x)=x^2+3 and g(x)=-2x+6. Find each of the following. See Example 1. (ƒ/g)(5)

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In Exercises 1–30, find the domain of each function. f(x) = 1/√(x - 3)

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For the pair of functions defined, find (ƒ-g)(x).Give the domain of each. See Example 2. ƒ(x)=3x+4, g(x)=2x-6

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For the pair of functions defined, find (ƒ+g)(x).Give the domain of each. See Example 2. ƒ(x)=2x^2-3x, g(x)=x^2-x+3

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For the pair of functions defined, find (ƒg)(x). Give the domain of each. See Example 2. ƒ(x)=2x^2-3x, g(x)=x^2-x+3

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For the pair of functions defined, find (ƒ-g)(x).Give the domain of each. See Example 2. ƒ(x)=√(4x-1), g(x)=1/x

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Textbook Question

In Exercises 1–30, find the domain of each function. g(x) = √(x −2)/(x-5)

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In Exercises 1–30, find the domain of each function. f(x) = (2x+7)/(x^3 - 5x^2 - 4x+20)

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In Exercises 31–50, find f−g and determine the domain for each function. f(x) = 2x + 3, g(x) = x − 1

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In Exercises 31–50, find f/g and determine the domain for each function. f(x) = x -5, g(x) = 3x²

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Use the graph to evaluate each expression. See Example 3(a). (ƒ-g)(1)

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In Exercises 31–50, find fg and determine the domain for each function. f(x) = x -5, g(x) = 3x²

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Use the graph to evaluate each expression. See Example 3(a). (ƒg)(1)

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In Exercises 31–50, find fg and determine the domain for each function. f(x) = 3 − x², g(x) = x² + 2x − 17

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In Exercises 31–50, find f−g and determine the domain for each function. f(x) = 3 − x², g(x) = x² + 2x − 16

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In Exercises 31–50, find ƒ+g and determine the domain for each function. f(x) = 3 − x², g(x) = x² + 2x − 15

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In Exercises 31–50, find ƒ+g and determine the domain for each function. f(x) = √x, g(x) = x − 4

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In Exercises 31–50, find f/g and determine the domain for each function. f(x) = √x, g(x) = x − 4

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In Exercises 31–50, find ƒ-g and determine the domain for each function. f(x) = 2 + 1/x, g(x) = 1/x

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In Exercises 31–50, find fg and determine the domain for each function. f(x)= = (5x+1)/(x² - 9), g(x) = (4x -2)/(x² - 9)

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In Exercises 31–50, find f−g and determine the domain for each function. f(x)= = (5x+1)/(x² - 9), g(x) = (4x -2)/(x² - 9)

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For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4. ƒ(x)=6x+2

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In Exercises 31–50, find f/g and determine the domain for each function. f(x)= = 8x/(x - 2), g(x) = 6/(x+3)

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In Exercises 31–50, find f−g and determine the domain for each function. f(x) = √(x +4), g(x) = √(x − 1)

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In Exercises 31–50, find ƒ+g and determine the domain for each function. f(x) = √(x +4), g(x) = √(x − 1)

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In Exercises 31–50, find ƒ+g, f−g, fg, and f/g. Determine the domain for each function. f(x) = √(x -2), g(x) = √(2-x)

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In Exercises 31–50, find ƒ+g, f−g, fg, and f/g. Determine the domain for each function. f(x) = √(x -2), g(x) = √(2-x)

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For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4. ƒ(x)=1/x^2

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For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4. ƒ(x)=-x^2

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For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4. ƒ(x)=x^2+3x+1

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In Exercises 51–66, find a. (fog) (2) b. (go f) (2) f(x)=4x-3, g(x) = 5x² - 2

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Let ƒ(x)=2x-3 and g(x)=-x+3. Find each function value. See Example 5. (ƒ∘g)(4)

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In Exercises 51–66, find a. (fog) (x) b. (go f) (x) f(x) = x²+2, g(x) = x² – 2

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In Exercises 51–66, find a. (fog) (x) b. (go f) (x) f(x) = 4-x, g(x) = 2x² +x+5

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In Exercises 59-64, let f(x) = 2x - 5 g(x) = 4x - 1 h(x) = x² + x + 2. Evaluate the indicated function without finding an equation for the function. (fog) (0)

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In Exercises 51–66, find a. (fog) (x) b. (go f) (x). f(x) = √x, g(x) = x − 1

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Let ƒ(x)=2x-3 and g(x)=-x+3. Find each function value. See Example 5. (ƒ∘ƒ)(2)

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In Exercises 59-64, let f(x) = 2x - 5 g(x) = 4x - 1 h(x) = x² + x + 2. Evaluate the indicated function without finding an equation for the function. g (f[h (1)])

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In Exercises 67-74, find a. (fog) (x) b. the domain of f o g. f(x) = 2/(x+3), g(x) = 1/x

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In Exercises 67-74, find a. (fog) (x) b. the domain of f o g. f(x) = x/(x+1), g(x) = 4/x

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Given functions f and g, find (a)(ƒ∘g)(x) and its domain. See Examples 6 and 7. ƒ(x)=-6x+9, g(x)=5x+7

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Given functions f and g, find (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7. ƒ(x)=8x+12, g(x)=3x-1

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Given functions f and g, find (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7. ƒ(x)=√x, g(x)=x+3

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In Exercises 75-82, express the given function h as a composition of two functions ƒ and g so that h(x) = (fog) (x). h(x) = ∛(x² – 9)

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In Exercises 76–81, find the domain of each function. g(x) = 4/(x - 7)

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Given functions f and g, find (a)(ƒ∘g)(x) and its domain. See Examples 6 and 7. ƒ(x)=x+2, g(x)=x^4+x^2-4

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In Exercises 75-82, express the given function h as a composition of two functions ƒ and g so that h(x) = (fog) (x). h(x) = |2x-5|

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Given functions f and g, find (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7. ƒ(x)=2/x, g(x)=x+1

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Given functions f and g, find (a)(ƒ∘g)(x) and its domain. See Examples 6 and 7. ƒ(x)=2/x, g(x)=x+1

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Given functions f and g, find (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7. ƒ(x)=√x, g(x)=1/(x+5)

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Use the graphs of f and g to solve Exercises 83–90.

Use the graphs of f and g to solve Exercises 83–90.

In Exercises 89–90, express the given function h as a composition of two functions f and g so that h(x) = (f ○ g)(x). h(x) = (x^2 + 2x - 1)^4

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Use the graphs of f and g to solve Exercises 83–90.

In Exercises 91–94, use the graphs of f and g to evaluate each composite function.

Let ƒ(x) = 3x^2 - 4 and g(x) = x^2 - 3x -4. Find each of the following. (f+g)(2k)

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Let ƒ(x) = √(x-2) and g(x) = x^2. Find each of the following, if possible. (ƒ ○ g)(x)

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Let ƒ(x) = √(x-2) and g(x) = x^2. Find each of the following, if possible. (f ○ g)(-6)

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Use the table to evaluate each expression, if possible. (f-g)(3)

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The graphs of two functions ƒ and g are shown in the figures.

Fill in the blank to correctly complete each sentence. The point (-1, 3) lies in quadrant ________ in the rectangular coordinate system.

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Fill in the blank to correctly complete each sentence. The point (4,_____ ) lies on the graph of the equation y = 3x - 6.

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Fill in the blank to correctly complete each sentence. The y-intercept of the graph of y = -2x + 6 is ________.

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Determine whether each statement is true or false. If false, explain why. The graph of y = x^2 + 2 has no x-intercepts.

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Determine whether each statement is true or false. If false, explain why. The midpoint of the segment joining (0, 0) and (4, 4) is 2.

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For the points P and Q, find (a) the distance d(P, Q) and (b) the coordinates of the mid-point M of line segment PQ. See Examples 2 and 5(a). P(-5,-6), Q(7,-1)

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For the points P and Q, find (a) the distance d(P, Q) and (b) the coordinates of the mid-point M of line segment PQ. See Examples 2 and 5(a). P(8,2), Q(3,5)

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For the points P and Q, find (a) the distance d(P, Q) and (b) the coordinates of the mid-point M of line segment PQ. See Examples 2 and 5(a). P(6,-2), Q(4,6)

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Determine whether the three points are the vertices of a right triangle. See Example 3. (-2,-8),(0,-4),(-4,-7)

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Determine whether the three points are the vertices of a right triangle. See Example 3. (-4,1),(1,4),(-6,-1)

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Determine whether the three points are the vertices of a right triangle. See Example 3. (-2,-5),(1,7),(3,15)

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Determine whether the three points are collinear. See Example 4. (0,-7),(-3,5),(2,-15)

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Determine whether the three points are collinear. See Example 4. (0,9),(-3,-7),(2,-19)

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Determine whether the three points are collinear. See Example 4. (-7,4),(6,-2),(-1,1)

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Find the coordinates of the other endpoint of each line segment, given its midpoint and one endpoint. See Example 5(b). midpoint (5, 8), endpoint (13, 10)

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Find the coordinates of the other endpoint of each line segment, given its midpoint and one endpoint. See Example 5(b). midpoint (12, 6), endpoint (19, 16)

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Find the coordinates of the other endpoint of each line segment, given its midpoint and one endpoint. See Example 5(b). midpoint (6a, 6b), endpoint (3a, 5b)

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Fill in the blank(s) to correctly complete each sentence. The domain of the relation { (3,5), (4, 9), (10, 13) } is _____.

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Fill in the blank(s) to correctly complete each sentence. The equation y = 4x - 6 defines a function with independent variable______ and dependent variable ________ .

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Fill in the blank(s) to correctly complete each sentence. For the function ƒ(x) = -4x + 2, ƒ(-2)= ______.

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Determine whether each relation defines a function. See Example 1. {(5,1),(3,2),(4,9),(7,8)}

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Determine whether each relation defines a function. See Example 1. {(8,0),(5,7),(9,3),(3,8)}

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Determine whether each relation defines a function. See Example 1. {(9,-2),(-3,5),(9,1)}

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Determine whether each relation defines a function. See Example 1. {(2,4),(0,2),(2,6)}

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Determine whether each relation defines a function. See Example 1.

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Determine whether each relation defines a function. See Example 1.

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Determine whether each relation defines a function, and give the domain and range. See Examples 1–4. {(1,1),(1,-1),(0,0),(2,4),(2,-4)}

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Determine whether each relation defines a function, and give the domain and range. See Examples 1–4. {(2,5),(3,7),(3,9),(5,11)}

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Textbook Question

Determine whether each relation defines a function, and give the domain and range. See Examples 1–4.

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Determine whether each relation defines a function, and give the domain and range. See Examples 1–4.

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Determine whether each relation defines a function, and give the domain and range. See Examples 1–4.

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Determine whether each relation defines a function, and give the domain and range. See Examples 1–4.

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Determine whether each relation defines a function, and give the domain and range. See Examples 1–4.

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Determine whether each relation defines y as a function of x. Give the domain and range. See Example 5. y=-7/(x-5)

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Let ƒ(x)=-3x+4 and g(x)=-x^2+4x+1. Find each of the following. Simplify if necessary. See Example 6. ƒ(-3)

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Let ƒ(x)=-3x+4 and g(x)=-x^2+4x+1. Find each of the following. Simplify if necessary. See Example 6. g(-2)

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Let ƒ(x)=-3x+4 and g(x)=-x^2+4x+1. Find each of the following. Simplify if necessary. See Example 6. g(10)

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Let ƒ(x)=-3x+4 and g(x)=-x^2+4x+1. Find each of the following. Simplify if necessary. See Example 6. ƒ(-7/3)

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Let ƒ(x)=-3x+4 and g(x)=-x^2+4x+1. Find each of the following. Simplify if necessary. See Example 6. g(1/2)

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Let ƒ(x)=-3x+4 and g(x)=-x^2+4x+1. Find each of the following. Simplify if necessary. See Example 6. g(-1/4)

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Let ƒ(x)=-3x+4 and g(x)=-x^2+4x+1. Find each of the following. Simplify if necessary. See Example 6. ƒ(p)

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Let ƒ(x)=-3x+4 and g(x)=-x^2+4x+1. Find each of the following. Simplify if necessary. See Example 6. g(k)

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Let ƒ(x)=-3x+4 and g(x)=-x^2+4x+1. Find each of the following. Simplify if necessary. See Example 6. g(-x)

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Let ƒ(x)=-3x+4 and g(x)=-x^2+4x+1. Find each of the following. Simplify if necessary. See Example 6. ƒ(x+2)

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Let ƒ(x)=-3x+4 and g(x)=-x^2+4x+1. Find each of the following. Simplify if necessary. See Example 6. ƒ(a+4)

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Let ƒ(x)=-3x+4 and g(x)=-x^2+4x+1. Find each of the following. Simplify if necessary. See Example 6. ƒ(2m-3)

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For each function, find (a) ƒ(2) and (b) ƒ(-1).See Example 7. ƒ = {(2,5),(3,9),(-1,11),(5,3)}

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For each function, find (a) ƒ(2) and (b) ƒ(-1).See Example 7. ƒ = {(-1,3),(4,7),(0,6),(2,2)}

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An equation that defines y as a function of x is given. (b) Find ƒ(3). x-4y=8

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An equation that defines y as a function of x is given. (b) Find ƒ(3). y+2x^2=3-x

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Find the value of the function for the given value of x. See Example 3. ƒ(x)=[[0.5x]], for x=7

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Find the value of the function for the given value of x. See Example 3. ƒ(x)=-[[-x]], for x=2.5

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Find the value of the function for the given value of x. See Example 3. ƒ(x)=2-[[-x]], for x=3.7

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Find the value of the function for the given value of x. See Example 3. ƒ(x)=[[x/4]], for x=7

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Find the value of the function for the given value of x. See Example 3. ƒ(x)=[[3-(x/2)]], for x=1

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Find the value of the function for the given value of x. See Example 3. ƒ(x)=[[x]], for x=-√2

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Determine whether each function is even, odd, or neither. See Example 5. ƒ(x)=-x^3+2x

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Determine whether each function is even, odd, or neither. See Example 5. ƒ(x)=x^5-2x^3

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Determine whether each function is even, odd, or neither. See Example 5. ƒ(x)=0.5x^4-2x^2+6

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Determine whether each function is even, odd, or neither. See Example 5. ƒ(x)=x^4-5x+8

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Determine whether each function is even, odd, or neither. See Example 5. ƒ(x)=x+1/x^5

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Determine whether each function is even, odd, or neither. See Example 5. ƒ(x)=x^4+4/x^2

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Determine whether each equation defines y as a function of x. x = (1/3)(y^2)

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Determine whether each equation defines y as a function of x. y = ±√(x-2)

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Let ƒ(x) = -2x^2 + 3x -6. Find each of the following. ƒ(-0.5)

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Determine whether each equation has a graph that is symmetric with respect to the x-axis, the y-axis, the origin, or none of these. 5y^2 + 5x^2 =30

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Consider the following nonlinear system. Work Exercises 75 –80 in order. y = | x - 1 | y = x^2 - 4 Use the definition of absolute value to write y = | x - 1 | as a piecewise-defined function.

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For each equation, (a) give a table with at least three ordered pairs that are solutions, and (b) graph the equation. See Examples 7 and 8. 2x+3y=5

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For each equation, (a) give a table with at least three ordered pairs that are solutions, and (b) graph the equation. See Examples 7 and 8. y=-x^2

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For each equation, (a) give a table with at least three ordered pairs that are solutions, and (b) graph the equation. See Examples 7 and 8. y=x^2

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For each equation, (a) give a table with at least three ordered pairs that are solutions, and (b) graph the equation. See Examples 7 and 8. y=|x+4|

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For each graph, determine whether y is a function of x. Give the domain and range of each relation.

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Determine whether each relation defines a function, and give the domain and range. See Examples 1–4.

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Determine whether each relation defines a function, and give the domain and range. See Examples 1–4.

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Determine whether each relation defines a function, and give the domain and range. See Examples 1–4.

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Determine whether each relation defines y as a function of x. Give the domain and range. See Example 5. x=y^4

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Determine whether each relation defines y as a function of x. Give the domain and range. See Example 5. y=-6x+4

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Determine whether each relation defines y as a function of x. Give the domain and range. See Example 5. x-y<4

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Determine whether each relation defines y as a function of x. Give the domain and range. See Example 5. y=-√x

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Determine whether each relation defines y as a function of x. Give the domain and range. See Example 5. y=√(7-2x)

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Determine whether each relation defines y as a function of x. Give the domain and range. See Example 5. y=2/(x-3)

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Determine the largest open intervals of the domain over which each function is (a) increasing. See Example 9.

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Determine the largest open intervals of the domain over which each function is (c) constant. See Example 9.

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For each function graphed, give the minimum and maximum values of ƒ(x) and the x-values at which they occur.

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For each function graphed, give the minimum and maximum values of ƒ(x) and the x-values at which they occur.

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To answer each question, refer to the following basic graphs. Which one is the graph of ƒ(x)=x^2? What is its domain?

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To answer each question, refer to the following basic graphs. Which one is the graph of ƒ(x)=x^3? What is its range?

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To answer each question, refer to the following basic graphs. Which one is the graph of ƒ(x)=|x|? What is the function value when x=1.5?

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To answer each question, refer to the following basic graphs. Which one is the graph of ƒ(x)=∛x? Is there any open interval over which the function is decreasing?

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To answer each question, refer to the following basic graphs. Which one is the graph of ƒ(x)=√x? What is its domain?

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Determine the intervals of the domain over which each function is continuous. See Example 1.

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Textbook Question

Determine the intervals of the domain over which each function is continuous. See Example 1.

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Textbook Question

Determine the intervals of the domain over which each function is continuous. See Example 1.

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Textbook Question

Determine the intervals of the domain over which each function is continuous. See Example 1.

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Textbook Question

Determine the intervals of the domain over which each function is continuous. See Example 1.

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Textbook Question

Graph each piecewise-defined function. See Example 2. ƒ(x)={x-1 if x≤3, 2 if x>3

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Textbook Question

Graph each piecewise-defined function. See Example 2. ƒ(x)={4-x if x<2, 1+2x if x≥2

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Textbook Question

Graph each piecewise-defined function. See Example 2. ƒ(x)={2x+1 if x≥0, x if x<0

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Textbook Question

Graph each piecewise-defined function. See Example 2. ƒ(x)={-3 if x≤1, -1 if x>1

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Textbook Question

Graph each piecewise-defined function. See Example 2. ƒ(x)={-2x if x<-3, 3x-1 if -3≤x≤2, -4x if x>2

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Textbook Question

Graph each piecewise-defined function. See Example 2. ƒ(x)={x^3+5 if x≤0, -x^2 if x<0

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Textbook Question

Graph each piecewise-defined function. See Example 2. ƒ(x)={-(1/2)x^2+2 if x≤2, (1/2)x if x>2

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Textbook Question

Give a rule for each piecewise-defined function. Also give the domain and range.

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Textbook Question

Give a rule for each piecewise-defined function. Also give the domain and range.

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Textbook Question

Give a rule for each piecewise-defined function. Also give the domain and range.

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Textbook Question

Find the value of the function for the given value of x. See Example 3. ƒ(x)={5 if 02, for x=5.6

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Textbook Question

Find the value of the function for the given value of x. See Example 3. ƒ(x)={3 if 04, for x=6.2

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Textbook Question

Graph each function. Give the domain and range. See Example 3. ƒ(x)=-[[x]]

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Textbook Question

Graph each function. Give the domain and range. See Example 3. g(x)=[[2x-1]]

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Textbook Question

Solve each problem. See Example 4. Suppose that the cost of mailing a letter weighing x ounces, where x>0, is ƒ(x)=55-15[[1-x]]cents. What is the cost for the first ounce?

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Textbook Question

Solve each problem. See Example 4. Suppose that the cost of mailing a letter weighing x ounces, where x>0, is ƒ(x)=55-15[[1-x]]cents. What is the cost of mailing a 2.6-ounce letter?

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Textbook Question

For each graph, determine whether y is a function of x. Give the domain and range of each relation.

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Textbook Question

For each graph, determine whether y is a function of x. Give the domain and range of each relation.

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Textbook Question

For each graph, determine whether y is a function of x. Give the domain and range of each relation.

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Textbook Question

For each graph, determine whether y is a function of x. Give the domain and range of each relation.

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Textbook Question

Use a graphing calculator to graph each equation in the standard viewing window. 3x + 4y = 6

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Textbook Question

Graph each equation. 2x +5y = 20

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Textbook Question

Graph each equation. 3y = x

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Textbook Question

Graph each equation. ƒ(x) = x

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Textbook Question

Use a graphing calculator to graph each equation in the standard viewing window. -2x + 5y = 10

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Textbook Question

Graph each equation. x - 4y = 8

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Textbook Question

Graph each equation. x = -5

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Textbook Question

Graph each equation. ƒ(x) = 3

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Textbook Question

Graph each function. ƒ(x) = -|x|

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Textbook Question

Graph each function. ƒ(x) = |x| -3

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Textbook Question

Graph each function. ƒ(x) = 2∛(x+1)-2

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Textbook Question

Graph each function. ƒ(x) = {|x| if x < 3 , 6-x if x ≥ 3}

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Textbook Question

Determine whether each equation has a graph that is symmetric with respect to the x-axis, the y-axis, the origin, or none of these. y^3 = x + 4

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Textbook Question

Determine whether each equation has a graph that is symmetric with respect to the x-axis, the y-axis, the origin, or none of these. |x| = |y|

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Textbook Question

In Exercises 39–50, graph the given functions, f and g, in the same rectangular coordinate system. Select integers for x, starting with -2 and ending with 2. Once you have obtained your graphs, describe how the graph of g is related to the graph of f. f(x) = x, g(x) = x + 3

In Exercises 65–70, use the graph of f to find each indicated function value. f(-3)

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Textbook Question

In Exercises 65–70, use the graph of f to find each indicated function value. f(4)

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Textbook Question

In Exercises 65–70, use the graph of f to find each indicated function value.

In Exercises 77–92, use the graph to determine a.the x-intercepts, if any; b. the y-intercept, if any; and e. the missing function values, indicated by question marks, below each graph.

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Textbook Question

In Exercises 77–92, use the graph to determine a.the x-intercepts, if any; b. the y-intercept, if any; and e. the missing function values, indicated by question marks, below each graph.

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Textbook Question

In Exercises 77–92, use the graph to determine a. the function's domain; b. the function's range; and e. the missing function values, indicated by question marks, below each graph.

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Textbook Question

In Exercises 49–56, identify each equation without completing the square. 4x^2 + 4y^2 + 12x + 4y + 1 = 0

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Textbook Question

In Exercises 53–64, complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. x² + y²+3x+5y+9/4=0

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Textbook Question

In Exercises 53–64, complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. x² + y² − x + 2y + 1 = 0

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Textbook Question

In Exercises 53–64, complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. x² + y² - 6y -7=0

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Textbook Question

In Exercises 53–64, complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. x² - 2x + y² – 15 = 0

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Textbook Question

In Exercises 53–64, complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. x² + y²+8x-2y-8=0

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Textbook Question

In Exercises 53–64, complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. x² + y² – 10x – 6y – 30 = 0

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Textbook Question

In Exercises 53–64, complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. x² + y²+6x+2y+6 = 0

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Textbook Question

In Exercises 41–52, give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. (x + 1)² + y² = 25

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Textbook Question

In Exercises 41–52, give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. x² + (y − 1)² = 1

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Textbook Question

In Exercises 41–52, give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. (x + 2)² + (y - 2)² = 4

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Textbook Question

In Exercises 41–52, give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. (x+3)² + (y + 2)² = 4

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Textbook Question

In Exercises 41–52, give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. (x − 3)² + (y + 1)² = 36

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Textbook Question

In Exercises 41–52, give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. x² + y² = 16

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Textbook Question

In Exercises 31–40, write the standard form of the equation of the circle with the given center and radius. Center (-4, 0), r = 10

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Textbook Question

In Exercises 31–40, write the standard form of the equation of the circle with the given center and radius. Center (−3, −1), r = √3

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Textbook Question

In Exercises 31–40, write the standard form of the equation of the circle with the given center and radius. Center (-1, 4), r = 2

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Textbook Question

In Exercises 31–40, write the standard form of the equation of the circle with the given center and radius. Center (3, 2), r = 5

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Textbook Question

In Exercises 31–40, write the standard form of the equation of the circle with the given center and radius. Center (0, 0), r = 7

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Textbook Question

In Exercises 19–30, find the midpoint of each line segment with the given endpoints. (√50, −6) and (√2, 6)

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Textbook Question

In Exercises 19–30, find the midpoint of each line segment with the given endpoints. (7√3, −6) and (3√3, −2)

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Textbook Question

In Exercises 19–30, find the midpoint of each line segment with the given endpoints. (8, 3√5) and (−6, 7√5)

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Textbook Question

In Exercises 19–30, find the midpoint of each line segment with the given endpoints. (-3, -4) and (6, −8)

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Textbook Question

In Exercises 65-66, a line segment through the center of each circle intersects the circle at the points shown. a. Find the coordinates of the circle's center. b. Find the radius of the circle. c. Use your answers from parts (a) and (b) to write the standard form of the circle's equation.

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Textbook Question

In Exercises 1–18, find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places. (-1/4, -1/7) and (3/4, 6/7)

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Textbook Question

In Exercises 1–18, find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places. (7/3, 1/5) and (1/3, 6/5)

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Textbook Question

In Exercises 1–18, find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places. (3√3, √5) and (−√3, 4√5)

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Textbook Question

In Exercises 1–18, find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places. (0, -√2) and (√7,0)

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Textbook Question

In Exercises 1–18, find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places. (0, −√3) and (√5, 0)

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Textbook Question

In Exercises 1–18, find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places. (3.5, 8.2) and (-0.5, 6.2)

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Textbook Question

In Exercises 1–18, find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places. (-2, -6) and (3, −4)

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Textbook Question

In Exercises 1–18, find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places. (0, 0) and (3,-4)

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Textbook Question

In Exercises 1–18, find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places. (4, -1) and (-6, 3)

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Textbook Question

In Exercises 1–18, find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places. (2, 3) and (14, 8)

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Textbook Question

In Exercises 19–30, find the midpoint of each line segment with the given endpoints. (-2, -8) and (−6, −2)

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Textbook Question

In Exercises 19–30, find the midpoint of each line segment with the given endpoints. (6, 8) and (2, 4)

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Textbook Question

Exercises 103–105 will help you prepare for the material covered in the next section. Solve by completing the square: y² – 6y — 4 = 0.

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Textbook Question

Exercises 103–105 will help you prepare for the material covered in the next section. Use a rectangular coordinate system to graph the circle with center (1, -1) and radius 1.

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Textbook Question

Exercises 103–105 will help you prepare for the material covered in the next section. Let (x1, y₁) = (7, 2) and (x2, y2) = (1, −1). Find √[(x2 − x1)² + (y2 − y₁)²]. Express the - answer in simplified radical form.

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Textbook Question

Fill in the blank(s) to correctly complete each sentence. The circle with center (3, 6) and radius 4 has equation _________.

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Textbook Question

In Exercises 109–111, give the center and radius of each circle. x^2 + y^2 - 4x + 2y - 4 = 0

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Textbook Question

In Exercises 107–108, write the standard form of the equation of the circle with the given center and radius. Center (-2. 4), r = 6

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Textbook Question

In Exercises 105–106, find the midpoint of each line segment with the given endpoints. (2, 6) and (-12, 4)

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Textbook Question

Find the given distances between points P, Q, R, and S on a number line, with coordi-nates -4, -1, 8, and 12, respectively. d(P, Q)

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Textbook Question

Find the given distances between points P, Q, R, and S on a number line, with coordi-nates -4, -1, 8, and 12, respectively. d(Q,R)

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Textbook Question

Fill in the blank(s) to correctly complete each sentence. The circle with equation x^2+y^2=49 has center with coordinates________ and radius equal to__________ .

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Textbook Question

In the following exercises, (a) find the center-radius form of the equation of each circle described, and (b) graph it. See Examples 1 and 2. center (0, 0), radius 6

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Textbook Question

In the following exercises, (a) find the center-radius form of the equation of each circle described, and (b) graph it. See Examples 1 and 2. center (2, 0), radius 6

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Textbook Question

In the following exercises, (a) find the center-radius form of the equation of each circle described, and (b) graph it. See Examples 1 and 2. center (0, 4), radius 4

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Textbook Question

In the following exercises, (a) find the center-radius form of the equation of each circle described, and (b) graph it. See Examples 1 and 2. center (5, -4), radius 7

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Textbook Question

In the following exercises, (a) find the center-radius form of the equation of each circle described, and (b) graph it. See Examples 1 and 2. center (-2, 5), radius 4

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Textbook Question

In the following exercises, (a) find the center-radius form of the equation of each circle described, and (b) graph it. See Examples 1 and 2. center (√2, √2), radius √2

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Textbook Question

Use each graph to determine an equation of the circle in (a) center-radius form and (b) general form.

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Textbook Question

Use each graph to determine an equation of the circle in (a) center-radius form and (b) general form.

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Textbook Question

Give the center and radius of the circle represented by each equation. See Examples 3 and 4. x^2+y^2+6x+8y+9=0

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Textbook Question

Give the center and radius of the circle represented by each equation. See Examples 3 and 4. x^2+y^2-4x+12y=-4

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Textbook Question

Describe the graph of each equation as a circle, a point, or nonexistent. If it is a circle, give the center and radius. If it is a point, give the coordinates. See Examples 3–5. x^2+y^2+4x-8y+32=0

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Textbook Question

Describe the graph of each equation as a circle, a point, or nonexistent. If it is a circle, give the center and radius. If it is a point, give the coordinates. See Examples 3–5. x^2+y^2+4x+14y=-54

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Textbook Question

Describe the graph of each equation as a circle, a point, or nonexistent. If it is a circle, give the center and radius. If it is a point, give the coordinates. See Examples 3–5. x^2+y^2+2x-6y+14=0

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Textbook Question

Describe the graph of each equation as a circle, a point, or nonexistent. If it is a circle, give the center and radius. If it is a point, give the coordinates. See Examples 3–5. x^2+y^2+4x+4y+8=0

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Textbook Question

Describe the graph of each equation as a circle, a point, or nonexistent. If it is a circle, give the center and radius. If it is a point, give the coordinates. See Examples 3–5. x^2+y^2-2x+12y-12=0

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Textbook Question

Work each of the following. Find the equation of a circle with center at (-4, 3), passing through the point (5, 8).Write it in center-radius form.

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Textbook Question

Find the distance between each pair of points, and give the coordinates of the midpoint of the line segment joining them. P(3, -1), Q(-4, 5)

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Textbook Question

Find the distance between each pair of points, and give the coordinates of the midpoint of the line segment joining them. M((-8, 2), N(3, -7)

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Textbook Question

Find the distance between each pair of points, and give the coordinates of the midpoint of the line segment joining them. A(-6, 3), B(-6,8)

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Textbook Question

In Exercises 67–70, graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. x² + y² = 16, x-y = 4

In Exercises 19–30, find the midpoint of each line segment with the given endpoints. (-7/2, 3/2) and (-5/2, -11/2)

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