Question
upstudy study bank question image url

For each pair of functions \( f \) and \( g \) below, find \( f(g(x)) \) and \( g(f(x)) \). Then, determine whether \( f \) and \( g \) are inverses of each other. Simplify your answers as much as possible. (Assume that your expressions are defined for all \( x \) in the domain of the composition You do not have to indicate the domain.) (a) \[ \begin{array}{l} f(x)=-\frac{3}{x}, x \neq 0 \\ g(x)=-\frac{3}{x}, x=0 \\ f(g(x))=\square \\ g(f(x))=\square \end{array} \] \( f \) and \( g \) are inverses of each other fand \( g \) are not inverses of each other (b) \( f(x)=-x+4 \) \[ \begin{array}{l} g(x)=x+4 \\ f(g(x))=\square \\ g(f(x))=\square \end{array} \] \( f \) and \( g \) are inverses of each other \( f \) and \( g \) are not inverses of each other

Ask by Allan Ball. in the United States
Mar 14,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

(a) \[ f(g(x)) = x, \quad g(f(x)) = x \] \( f \) and \( g \) are inverses of each other. (b) \[ f(g(x)) = -x, \quad g(f(x)) = -x + 8 \] \( f \) and \( g \) are not inverses of each other.

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

Beyond the Answer

Let's tackle each pair of functions one by one! (a) Here, we have \( f(x) = -\frac{3}{x} \) for \( x \neq 0 \) and \( g(x) = -\frac{3}{x} \) for \( x = 0 \). However, since \( g(x) \) is not defined for \( x \neq 0 \), we actually only need to consider \( f(g(x)) \) for cases where \( g(x) \) is defined. 1. \( f(g(x)) = f(-\frac{3}{0}) \) which is undefined since we cannot divide by zero. 2. \( g(f(x)) = g(-\frac{3}{x}) = -\frac{3}{-\frac{3}{x}} = x \) for \( x \neq 0 \). Since \( f(g(x)) \) is undefined, they cannot be inverses of each other. (b) Now, let's look at the second pair \( f(x) = -x + 4 \) and \( g(x) = x + 4 \). 1. For \( f(g(x)) = f(x + 4) = -(x + 4) + 4 = -x - 4 + 4 = -x \). 2. For \( g(f(x)) = g(-x + 4) = (-x + 4) + 4 = -x + 8 \). Since \( f(g(x)) = -x \) and \( g(f(x)) \) does not yield \( x \) (but rather \( -x + 8 \)), we can conclude that \( f \) and \( g \) are not inverses of each other.

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy