Question

For the function $f(x)=x^{3}, f(2)=2^{3}=8$. Explain why $f^{-1}(8)=2 .$

Name: Problem 98, Chapter 9: Intermediate Algebra for College Students
Uploaded: 2020-05-08T15:57:35-08:00
Duration: 31 s
Channel: Linh Vu
Description: For the function f(x)=x^3, f(2)=2^3=8. Explain why f^-1(8)=2 .

   For the function $f(x)=x^{3}, f(2)=2^{3}=8$. Explain why $f^{-1}(8)=2 .$

Intermediate Algebra for College Students

Allen R. Angel;… 10th Edition

Chapter 9, Problem 98

Instant Answer

Solved by Expert Linh Vu

05/08/2020

Step 1

We know that f(x) = x^3, and we are given that f(2) = 8. Show more…

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For the function $f(x)=x^{3}, f(2)=2^{3}=8$. Explain why $f^{-1}(8)=2 .$

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Key Concepts

Inverse Functions

An inverse function reverses the relationship of the original function by swapping its inputs and outputs. If f maps an element x to y (i.e., f(x)=y), then its inverse, f?¹, maps y back to x (i.e., f?¹(y)=x), provided the function is one-to-one.

One-to-One Functions

A function must be one-to-one (injective) to have an inverse, meaning every output is associated with exactly one input. This property ensures that the inverse function is uniquely defined, as is the case for the function f(x)=x³.

Function and Inverse Evaluation

Evaluating a function and its inverse involves confirming that f(a)=b implies f?¹(b)=a. For example, if f(2)=8 for the function f(x)=x³, by the definition of inverse functions, we deduce that f?¹(8)=2.

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