Textbook Question
[Technology Exercise]
a. Graph the functions f(x) = x/2 and g(x) = 1 + (4/x) together to identify the values of x for which
x/2 > 1 + 4/x
b. Confirm your findings in part (a) algebraically.
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Ch. 1 - Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 1, Problem 1.1.50
Even and Odd Functions
In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.
f(x) = x² + x
Verified step by step guidance1
To determine if a function is even, odd, or neither, we need to analyze the function's symmetry properties. A function f(x) is even if f(-x) = f(x) for all x in the domain, and it is odd if f(-x) = -f(x) for all x in the domain.
Start by substituting -x into the function f(x) = x² + x to find f(-x). This gives us f(-x) = (-x)² + (-x).
Simplify the expression for f(-x). Since (-x)² = x², we have f(-x) = x² - x.
Compare f(-x) = x² - x with the original function f(x) = x² + x. Notice that f(-x) is not equal to f(x), so the function is not even.
Next, check if f(-x) = -f(x). Calculate -f(x) = -(x² + x) = -x² - x. Since f(-x) = x² - x is not equal to -f(x) = -x² - x, the function is not odd. Therefore, the function is neither even nor odd.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Even Functions
A function is classified as even if it satisfies the condition f(-x) = f(x) for all x in its domain. This means that the graph of the function is symmetric with respect to the y-axis. A common example of an even function is f(x) = x², where substituting -x yields the same output as substituting x.
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Exponential Functions
Odd Functions
A function is considered odd if it meets the condition f(-x) = -f(x) for all x in its domain. This indicates that the graph of the function is symmetric with respect to the origin. An example of an odd function is f(x) = x³, where substituting -x results in the negative of the output for x.
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Neither Even Nor Odd Functions
A function is classified as neither even nor odd if it does not satisfy the conditions for either classification. This means that f(-x) does not equal f(x) and also does not equal -f(x). An example is f(x) = x² + x, as it does not exhibit symmetry about the y-axis or the origin.
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06:21Properties of Functions
Related Practice
Textbook Question
Finding a Viewing Window
In Exercises 5–30, find an appropriate graphing software viewing window for the given function and use it to display that function’s graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function.
y = x + (1/10) sin 30x
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Textbook Question
Using the Half-Angle Formulas
Find the function values in Exercises 47–50.
sin² 3π/8
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Textbook Question
Refer to the given figure. Write the radius r of the circle in terms of α and θ.
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Textbook Question
Graph the functions in Exercises 37–56.
y = |x − 2|
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Textbook Question
In Exercises 5–8, determine whether the graph of the function is symmetric about the 𝔂-axis, the origin, or neither.
𝔂 = e⁻ˣ²
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