Ch. 5 - Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 5 - Integration

Problem 5.4.7

Chapter 5, Problem 5.4.7

Is x¹² an even or odd function? Is sin x² an even or odd function?

Verified step by step guidance

Step 1: Recall the definitions of even and odd functions. An even function satisfies f(-x) = f(x), meaning it is symmetric about the y-axis. An odd function satisfies f(-x) = -f(x), meaning it is symmetric about the origin.

Step 2: Analyze the function x¹². Substitute -x into the function: f(-x) = (-x)¹². Since raising -x to an even power results in x¹² (because the negative sign disappears), f(-x) = f(x). Therefore, x¹² is an even function.

Step 3: Analyze the function sin(x²). Substitute -x into the function: f(-x) = sin((-x)²). Since squaring -x results in x² (because the negative sign disappears), f(-x) = sin(x²). Therefore, sin(x²) satisfies f(-x) = f(x), making it an even function.

Step 4: Summarize the results. Both x¹² and sin(x²) are even functions because they satisfy the condition f(-x) = f(x).

Step 5: Reflect on the symmetry of these functions. x¹² is symmetric about the y-axis due to the even power, and sin(x²) is symmetric about the y-axis because the argument x² is unaffected by the sign of x.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Even and Odd Functions

A function is classified as even if it satisfies the condition f(-x) = f(x) for all x in its domain, meaning its graph is symmetric about the y-axis. Conversely, a function is odd if it meets the condition f(-x) = -f(x), indicating that its graph is symmetric about the origin. Understanding these definitions is crucial for determining the nature of specific functions.

Polynomial Functions

Polynomial functions, such as x¹², are expressions that involve variables raised to whole number powers. The degree of the polynomial and the coefficients determine its behavior. In the case of even-degree polynomials, they are always even functions, as substituting -x yields the same result as substituting x.

Trigonometric Functions

Trigonometric functions, like sin(x), have specific properties regarding symmetry. The sine function is an odd function, meaning sin(-x) = -sin(x). When considering sin(x²), we must analyze the argument x², which is always non-negative, affecting the overall symmetry and classification of the function.

Recommended video:

6:04

Introduction to Trigonometric Functions

Related Practice

Textbook Question

Symmetry of composite functions Prove that the integrand is either even or odd. Then give the value of the integral or show how it can be simplified. Assume f and g are even functions and p and q are odd functions.

∫ᵃ₋ₐ ƒ(p(𝓍)) d𝓍

110

views

Textbook Question

Cubic zero net area Consider the graph of the cubic y = 𝓍 (𝓍― a) (𝓍― b), where 0 < a < b. Verify that the graph bounds a region above the 𝓍-axis, for 0 < 𝓍 < a , and bounds a region below the 𝓍-axis, for a < 𝓍 < b. What is the relationship between a and b if the areas of these two regions are equal?

views

Textbook Question

Let ƒ(𝓍) = c, where c is a positive constant. Explain why an area function of ƒ is an increasing function.

views

Textbook Question

Areas of regions Find the area of the region bounded by the graph of ƒ and the 𝓍-axis on the given interval.

ƒ(𝓍) = sin 𝓍 on [―π/4, 3π/4]

108

views

Textbook Question

Symmetry in integrals Use symmetry to evaluate the following integrals.

∫²₋₂ (x² + x³) dx