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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2:37 minutes

Problem 1.6

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

Verified step by step guidance

To determine symmetry about the y-axis, check if replacing x with -x in the function yields the same function. For the given function y = x^(2/5), replace x with -x to get y = (-x)^(2/5).

Simplify the expression (-x)^(2/5). Since the exponent 2/5 is a positive rational number, (-x)^(2/5) simplifies to x^(2/5) because the square of a negative number is positive.

Since y = (-x)^(2/5) simplifies to y = x^(2/5), the function is symmetric about the y-axis.

To check for symmetry about the origin, replace both x with -x and y with -y in the original equation. This gives -y = (-x)^(2/5).

Since -y = x^(2/5) does not simplify to the original equation y = x^(2/5), the function is not symmetric about the origin. Therefore, the graph of the function is symmetric about the y-axis only.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Symmetry about the y-axis

A function is symmetric about the y-axis if replacing x with -x in the function yields the same output. Mathematically, this means that f(-x) = f(x) for all x in the domain of the function. This type of symmetry indicates that the graph of the function is a mirror image across the y-axis.

Symmetry about the origin

A function is symmetric about the origin if replacing x with -x and y with -y results in the same function. This is expressed as f(-x) = -f(x). Functions with this symmetry exhibit rotational symmetry of 180 degrees around the origin, meaning that if you rotate the graph, it looks the same.

Analyzing function symmetry

To determine the symmetry of a function, one can evaluate the function at both x and -x. By comparing the results, one can conclude whether the function is symmetric about the y-axis, the origin, or neither. This analysis is crucial for understanding the behavior of the graph and its visual representation.

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Related Practice

Textbook Question

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

𝔂 = x⁴ + 1

x³ - 2x

Textbook Question

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

𝔂 = x - sin x

Textbook Question

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

𝔂 = x cos x

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 - 2x - 1

Textbook Question

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

𝔂 = 1 - cos x

Textbook Question

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

𝔂 = x² + 1

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|

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In Exercises 5–8, determine whether the graph of the function is symmetric about the 𝔂-axis, the origin, or neither.𝔂 = x²/⁵

Key Concepts

Symmetry about the y-axis

Symmetry about the origin

Analyzing function symmetry

Watch next

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

𝔂 = x²/⁵