/*! 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 22 Verify that \(f\) and \(g\) are ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 22

Verify that \(f\) and \(g\) are inverse functions (a) algebraically and (b) graphically. $$f(x)=x-5, \quad g(x)=x+5$$

Short Answer

Expert verified
Yes, based on the results of the algebraic and graphical verification, the functions \(f(x)=x-5\) and \(g(x)=x+5\) are indeed inverse functions of each other.

Step by step solution

01

Algebraic Verification

It needs to be shown that \(f(g(x))=x\) and \(g(f(x))=x\). Starting with \(f(g(x))\), substitute \(g(x)\) into \(f(x)\): \(f(g(x))=f(x+5)=(x+5)-5=x\). Similarly for \(g(f(x))\), substitute \(f(x)\) into \(g(x)\): \(g(f(x))=g(x-5)=(x-5)+5=x\). Hence, both \(f(g(x))\) and \(g(f(x))\) equal to \(x\), confirming that the functions are inverses of each other algebraically.
02

Graphical Verification

Next step is to visually verify that \(f(x)\) and \(g(x)\) are inverse functions. This is done by graphing both the functions and checking if one is the mirror image of the other, reflected over the line \(y=x\). The graph of \(f(x)=x-5\) is a straight line, sloping upwards with y-intercept of -5. The graph of \(g(x)=x+5\) is also a straight line, sloping upwards but with y-intercept of 5. These two graphs are reflected over the line \(y=x\), which means they are inverse functions graphically.

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!

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

Sketch the graph of the function. $$f(x)=\left\\{\begin{array}{ll}x^{2}+5, & x \leq 1 \\\\-x^{2}+4 x+3, & x>1\end{array}\right.$$

The diameter of the largest particle that can be moved by a stream varies approximately directly as the square of the velocity of the stream. A stream with a velocity of \(\frac{1}{4}\) mile per hour can move coarse sand particles about 0.02 inch in diameter. Approximate the velocity required to carry particles 0.12 inch in diameter.

Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$k(x)=1 /(x-3)$$

(a) use a graphing utility to graph the function and (b) state the domain and range of the function. $$k(x)=4\left(\frac{1}{2} x-\left[\left[\frac{1}{2} x\right]\right]\right)^{2}$$

Find the difference quotient and simplify your Answer: $$f(x)=x^{2 / 3}+1, \quad \frac{f(x)-f(8)}{x-8}, \quad x \neq 8$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

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.