/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 75 Determine whether the function i... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 75

Determine whether the function is even, odd, or neither. $$f(x)=\sin x \cos x$$

Short Answer

Expert verified
The function is odd.

Step by step solution

01

Define Properties of Even and Odd Functions

For a function to be even, it must satisfy the condition \( f(-x) = f(x) \) for all \( x \). For a function to be odd, the condition \( f(-x) = -f(x) \) must hold true for all \( x \). If neither condition is met, the function is neither even nor odd.
02

Substitute -x into the Function

Substitute \(-x\) into the function: \( f(-x) = \sin(-x) \cdot \cos(-x) \). Recall that \( \sin(-x) = -\sin(x) \) and \( \cos(-x) = \cos(x) \). Therefore, \( f(-x) = -\sin(x) \cdot \cos(x) \).
03

Compare \(f(-x)\) with \(f(x)\)

From step 2, we have \( f(-x) = -\sin(x) \cdot \cos(x) \) and the original function \( f(x) = \sin(x) \cdot \cos(x) \).We observe that\( f(-x) = -f(x) \).
04

Conclude the Nature of the Function

Since \( f(-x) = -f(x) \), the function \( f(x) = \sin(x) \cos(x) \) meets the condition for an odd function. Therefore, the function is odd.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Even and Odd Functions
Understanding whether a function is even, odd, or neither is essential in precalculus. This concept helps in determining the symmetry of the function's graph. Here鈥檚 how you identify these functions:
  • Even Functions: A function is even if for every x, the equation \( f(-x) = f(x) \) holds true. This means that the graph of an even function is symmetrical about the y-axis.
  • Odd Functions: Conversely, a function is odd if for all x, \( f(-x) = -f(x) \) is satisfied. An odd function鈥檚 graph has rotational symmetry about the origin.
  • Neither: If a function does not satisfy either condition, it is considered neither even nor odd.
The function in the exercise, \( f(x) = \sin(x)\cos(x) \), is found to be odd because replacing x with -x gives \( f(-x) = -f(x) \). This makes it meet the definition of an odd function.
Trigonometric Functions
Trigonometric functions like sine, cosine, and tangent are fundamental to mathematics, describing periodic phenomena.
  • Sine Function: Denoted as \( \sin(x) \), it is an odd function. This means that \( \sin(-x) = -\sin(x) \).
  • Cosine Function: Represented as \( \cos(x) \), it is an even function since \( \cos(-x) = \cos(x) \).
  • Tangent Function: Known as \( \tan(x) \), is an odd function, similar to sine.
When analyzing \( f(x) = \sin(x)\cos(x) \), notice that it combines both sine and cosine. The sine part adheres to its odd function property, creating a negative product when x is replaced by -x, which helps determine the odd nature of the entire function.
Function Properties
Function properties aid in understanding more about a function's behavior without graphing it. These properties extend beyond even and odd characteristics:
  • Periodicity: Many trigonometric functions, like sine and cosine, repeat their values after a certain interval. This is the function鈥檚 period. For instance, \( \sin(x) \) and \( \cos(x) \) have periods of \( 2\pi \).
  • Symmetry: Knowing if a function is even or odd can help quickly identify symmetry, which simplifies graphing and understanding the function鈥檚 graph.
  • Amplitude and Phase Shift: In trigonometric contexts, the amplitude indicates the height of function peaks, while phase shift describes horizontal shifts on a graph. Functions like \( \sin(x) \) often have these characteristics.
These properties describe how a function behaves and interacts, contributing to a clearer understanding of how it operates within its domain. Identifying even and odd properties in functions can simplify complex analysis, making it easier to predict and manipulate function behavior.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t, \cos t,\) and \(\tan t\). $$\left(-\frac{20}{29}, \frac{21}{29}\right)$$

(a) Graph the function and make a conjecture, and (b) prove that your conjecture is true. $$y=\tan ^{-1} x+\tan ^{-1} \frac{1}{x}$$

Determine whether the function is even, odd, or neither. $$f(x)=x \sin ^{3} x$$

Write the first expression in terms of the second if the terminal point determined by \(t\) is in the given quadrant. \(\sin t,\) sec \(t ; \quad\) Quadrant IV

Find the period and graph the function. $$y=2 \sec \left(\frac{1}{2} x-\frac{\pi}{3}\right)$$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.