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

Properties of Functions

Calculus

0. Functions

Properties of Functions

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2:30 minutes

Problem 1.9

Textbook Question

In Exercises 9–16, determine whether the function is even, odd, or neither.

𝔂 = x² + 1

Verified step by step guidance

To determine if a function is even, odd, or neither, we need to analyze its symmetry properties. A function y = 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 y = x² + 1 to find f(-x). This gives us f(-x) = (-x)² + 1.

Simplify the expression f(-x) = (-x)² + 1. Since (-x)² = x², we have f(-x) = x² + 1.

Compare f(-x) with f(x). We have f(-x) = x² + 1 and f(x) = x² + 1. Since f(-x) = f(x), the function is even.

Conclude that the function y = x² + 1 is even because it satisfies the condition 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 considered 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 substituting -x yields the same result as substituting x.

Odd Functions

A function is classified as 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 function's value.

Neither Even Nor Odd Functions

A function is neither even nor odd if it does not satisfy the conditions for either classification. This means that the function's graph lacks symmetry with respect to both the y-axis and the origin. For instance, the function f(x) = x² + 1 is neither even nor odd, as it does not exhibit the required symmetries.

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Related Practice

Textbook Question

Suppose that ƒ and g are both odd functions defined on the entire real line. Which of the following (where defined) are even? odd?

a. ƒg

b. ƒ³

c. ƒ(sin x)

d. g(sec x)

e. |g|

Textbook Question

In Exercises 5–8, determine whether the graph of the function is symmetric about the 𝔂-axis, the origin, or neither.

𝔂 = x²/⁵

Textbook Question

In Exercises 5–8, determine whether the graph of the function is symmetric about the 𝔂-axis, the origin, or neither.

𝔂 = x² - 2x - 1

Textbook Question

In Exercises 9–16, determine whether the function is even, odd, or neither.

𝔂 = 1 - cos x

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 = 1/|x|

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 = x³/8

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 = (−x)²/³

Textbook Question

Find the largest interval on which the given function is increasing.

b. ƒ(x) = (x + 1)⁴

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