Question
upstudy study bank question image url

Are the following functions inverse \( f(x)=\sqrt[3]{x-5} \) \( g(x)=x^{3}+5 \)

Ask by Reed Cross. in the United States
Mar 14,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

\( f \) and \( g \) are inverses of each other.

Solution

To verify that \( f \) and \( g \) are inverses, we must show that: 1. \( f(g(x)) = x \) 2. \( g(f(x)) = x \) **Step 1: Compute \( f(g(x)) \)** Given \[ g(x)=x^3+5, \] we substitute \( g(x) \) into \( f \): \[ f(g(x)) = f(x^3+5) = \sqrt[3]{(x^3+5)-5} = \sqrt[3]{x^3}. \] Since the cube root of \( x^3 \) is \( x \) for all real \( x \), we have: \[ f(g(x)) = x. \] **Step 2: Compute \( g(f(x)) \)** Given \[ f(x)=\sqrt[3]{x-5}, \] we substitute \( f(x) \) into \( g \): \[ g(f(x)) = g\left(\sqrt[3]{x-5}\right) = \left(\sqrt[3]{x-5}\right)^3 + 5. \] Since raising the cube root to the third power returns the original argument, \[ \left(\sqrt[3]{x-5}\right)^3 = x-5, \] we get: \[ g(f(x)) = (x-5) + 5 = x. \] Since both compositions yield \( x \), the functions \( f \) and \( g \) are indeed inverses of each other.

Answered by UpStudy AI and reviewed by a Professional Tutor

error msg
Explain
Simplify this solution

The Deep Dive

To determine if the functions \( f(x) = \sqrt[3]{x-5} \) and \( g(x) = x^3 + 5 \) are inverses, we need to check if \( f(g(x)) = x \) and \( g(f(x)) = x \). Starting with \( f(g(x)) \): \[ f(g(x)) = f(x^3 + 5) = \sqrt[3]{(x^3 + 5) - 5} = \sqrt[3]{x^3} = x \] Now, checking \( g(f(x)) \): \[ g(f(x)) = g(\sqrt[3]{x-5}) = (\sqrt[3]{x-5})^3 + 5 = x - 5 + 5 = x \] Since both compositions yield \( x \), the functions \( f \) and \( g \) are indeed inverses of each other! The relationship between a function and its inverse is a beautiful part of mathematics; it shows that each function undoes the other, reflecting the symmetry between operations. If you’re interested in how these functions fit into the broader landscape of functions, consider exploring concepts of transformations, where shifts, reflections, and stretches apply to parent functions. It expands your understanding of function behavior and their relationships!

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