Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

0. Functions

Common Functions

Calculus

0. Functions

Common Functions

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1:56 minutes

Problem 1.1.48

Textbook Question

Even and Odd Functions

In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.

f(x) = x⁻⁵

Verified step by step guidance

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.

Let's start by finding f(-x) for the given function f(x) = x⁻⁵. Substitute -x into the function: f(-x) = (-x)⁻⁵.

Simplify the expression f(-x) = (-x)⁻⁵. Since the exponent is an odd number, we have f(-x) = -x⁻⁵.

Now, compare f(-x) with f(x). We have f(-x) = -x⁻⁵ and f(x) = x⁻⁵. Notice that f(-x) = -f(x), which satisfies the condition for the function to be odd.

Therefore, the function f(x) = x⁻⁵ is an odd function because f(-x) = -f(x) for all x in its domain.

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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. For example, the function f(x) = x² is even because f(-x) = (-x)² = x².

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, as f(-x) = (-x)³ = -x³.

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) or -f(x) for all x in its domain. For instance, the function f(x) = x + 1 is neither even nor odd, as f(-x) = -x + 1 does not equal f(x) or -f(x).

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Related Practice

Textbook Question

Increasing and Decreasing Functions

Graph the functions in Exercises 37–46. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.

y = −x³

Textbook Question

The Greatest and Least Integer Functions

Does ⌊x⌋ = ⌈x⌉ for all real x? Give reasons for your answer.

Textbook Question

The Greatest and Least Integer Functions

For what values of x is

b. ⌈x⌉ = 0

Textbook Question

Can a function be both even and odd? Give reasons for your answer.

Textbook Question

Theory and Examples

The accompanying figure shows a rectangle inscribed in an isosceles right triangle whose hypotenuse is 2 units long.

a. Express the y-coordinate of P in terms of x. (You might start by writing an equation for the line AB.)

<IMAGE>

Textbook Question

Theory and Examples

In Exercises 69 and 70, match each equation with its graph. Do not use a graphing device, and give reasons for your answer.

a. y = x⁴

b. y = x⁷

c. y = x¹⁰

<IMAGE>

Textbook Question

Industrial costs A power plant sits next to a river where the river is 800 ft wide. Laying a new cable from the plant to a location in the city 2 mi downstream on the opposite side costs $180 per foot across the river and $100 per foot along the land.

<IMAGE>

b. Generate a table of values to determine whether the least expensive location for point Q is less than 2000 ft or greater than 2000 ft from point P.

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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