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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2:00 minutes

Problem 1.1.60

Textbook Question

Even and Odd Functions

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

sin x²

Verified step by step guidance

Understand the definitions: A function f(x) is even if f(-x) = f(x) for all x in the domain, and odd if f(-x) = -f(x) for all x in the domain.

Consider the function given: f(x) = sin(x²). We need to check the behavior of f(-x) compared to f(x).

Calculate f(-x): Substitute -x into the function to get f(-x) = sin((-x)²). Since (-x)² = x², this simplifies to sin(x²).

Compare f(-x) and f(x): We find that f(-x) = sin(x²) = f(x), which matches the condition for an even function.

Conclude: Since f(-x) = f(x), the function sin(x²) is even. It does not satisfy the condition for an odd function, which would require f(-x) = -f(x).

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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. Common examples include polynomial functions with only even powers, such as f(x) = 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. Typical examples include polynomial functions with only odd powers, such as f(x) = x³.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, have specific properties regarding their symmetry. The sine function is an odd function, meaning sin(-x) = -sin(x). Understanding these properties is crucial when analyzing functions like sin(x²) to determine their evenness or oddness.

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Related Practice

Textbook Question

Assume f is an odd function and that both f and g are one-to-one. Use the (incomplete) graph of f and the graph of g to find the following function values. <IMAGE>

f⁻¹( g⁻¹(4))

Textbook Question

Assume f is an odd function and that both f and g are one-to-one. Use the (incomplete) graph of f and the graph of g to find the following function values. <IMAGE>

f^-1(1 + f(-3))

Textbook Question

A pen in the shape of an isosceles right triangle with legs of length x ft and hypotenuse of length h ft is to be built. If fencing costs $5/ft for the legs and $10/ft for the hypotenuse, write the total cost C of construction as a function of h.

Textbook Question

Theory and Examples

The variables r and s are inversely proportional, and r = 6 when s = 4. Determine s when r = 10.

Textbook Question

Even and Odd Functions

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

h(t) = |t³|

Textbook Question

Even and Odd Functions

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

g(x) = x/(x² − 1)

Textbook Question

Even and Odd Functions

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

g(x) = x⁴ + 3x² − 1

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) = x² + x

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Even and Odd FunctionsIn Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.sin x²

Key Concepts

Even Functions

Odd Functions

Trigonometric Functions

Watch next

Even and Odd Functions

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

sin x²