Question

For each pair of inverse functions, show that $\left(f \circ f^{-1}\right)(x)=x$ and $\left(f^{-1} \circ f\right)(x)=x$. $f(x)=-\frac{1}{3} x+2, f^{-1}(x)=-3 x+6$

Name: Problem 80, Chapter 9: Intermediate Algebra for College Students
Uploaded: 2020-06-24T15:34:16-08:00
Duration: 45 s
Channel: James Kiss
Description: For each pair of inverse functions, show that (f ∘f^-1)(x)=x and (f^-1 ∘f)(x)=x. f(x)=-(1)/(3) x+2, f^-1(x)=-3 x+6

   For each pair of inverse functions, show that $\left(f \circ f^{-1}\right)(x)=x$ and $\left(f^{-1} \circ f\right)(x)=x$.
$f(x)=-\frac{1}{3} x+2, f^{-1}(x)=-3 x+6$

Intermediate Algebra for College Students

Allen R. Angel;… 10th Edition

Chapter 9, Problem 80

Instant Answer

Solved by Expert James Kiss

06/24/2020

Step 1

This means we need to find $f(f^{-1}(x))$. We are given $f^{-1}(x) = -3x + 6$. So, we need to find $f(-3x + 6)$. We are given $f(x) = -\frac{1}{3}x + 2$. So, we can substitute $-3x + 6$ for $x$ in this expression: $f(-3x + 6) = -\frac{1}{3}(-3x + 6) + Show more…

Show all steps

Thanks for your feedback!

For each pair of inverse functions, show that $\left(f \circ f^{-1}\right)(x)=x$ and $\left(f^{-1} \circ f\right)(x)=x$. $f(x)=-\frac{1}{3} x+2, f^{-1}(x)=-3 x+6$

Play audio

Feedback

James Kiss and 90 other educators are ready to help you.

Ask a new question

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Inverse Functions

An inverse function reverses the effect of the original function. If f is a function with an inverse f?¹, then applying f followed by f?¹ (or vice versa) returns the original input value. This concept is fundamental because it allows one to 'undo' an operation and is only valid if the function is one-to-one and onto.

Function Composition

Function composition involves applying one function to the result of another function. In the context of inverse functions, composing a function with its inverse in either order should yield the identity function, meaning that the input is returned unchanged.

Identity Function

The identity function is a key concept in function composition and inverse operations. It is the function that maps every element to itself, denoted typically by I(x)=x, and it is the result of composing a function with its inverse.

Please give Ace some feedback