Question
upstudy study bank question image url

28. Consider the equation \( 2 x-5=3 \). Let \( f \) be the function defined by the left side of the equation so that \( f(x)=2 x-5 \) and the equation can be written as \( f(x)=3 \). A. Explain why the solution of \( 2 x-5=3 \) is \( f^{-1}(3) \). B. Write an equation for \( f^{-1}(x) \). Then use the equation to solve \( 2 x-5=3 \).

Ask by Pollard Hobbs. in the United States
Mar 14,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

**A.** The solution to \( 2x - 5 = 3 \) is \( x = 4 \). **B.** The inverse function is \( f^{-1}(x) = \frac{x + 5}{2} \). Using this, the solution is \( x = 4 \).

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

Bonus Knowledge

To understand why the solution of \( 2x - 5 = 3 \) is \( f^{-1}(3) \), let's break it down. The function defined as \( f(x) = 2x - 5 \) describes a relationship between \( x \) and \( f(x) \). When you set \( f(x) = 3 \), you are effectively asking what input \( x \) renders the output 3. The inverse function \( f^{-1}(x) \) will give you that input directly, meaning \( f^{-1}(3) \) yields the value of \( x \) that solves the original equation. Now, to find the inverse function \( f^{-1}(x) \), start by replacing \( f(x) \) with \( y \): so we have \( y = 2x - 5 \). Next, solve for \( x \): 1. \( y + 5 = 2x \) 2. \( x = \frac{y + 5}{2} \) Thus, the inverse function can be expressed as \( f^{-1}(x) = \frac{x + 5}{2} \). To solve \( 2x - 5 = 3 \) using this, simply substitute 3 into \( f^{-1}(x) \): \[ f^{-1}(3) = \frac{3 + 5}{2} = \frac{8}{2} = 4 \] So, the solution to \( 2x - 5 = 3 \) is \( x = 4 \).

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