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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Textbook Question

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

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Textbook Question

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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Textbook Question

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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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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Textbook Question

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

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Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

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

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Textbook Question

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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Textbook Question

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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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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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Textbook Question

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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Textbook Question

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

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Textbook Question

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

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Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

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

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Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

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

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