Question

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

Name: Problem 84, Chapter 9: Intermediate Algebra for College Students
Uploaded: 2020-09-18T03:01:06-08:00
Duration: 1 min
Channel: Sujit Kumar
Description: For each pair of inverse functions, show that (f ∘f^-1)(x)=x and (f^-1 ∘f)(x)=x. f(x)=-(2)/(x^'), f^-1(x)=-(2)/(x)

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

Intermediate Algebra for College Students

Allen R. Angel;… 10th Edition

Chapter 9, Problem 84

Instant Answer

Solved by Expert Sujit Kumar

09/18/2020

Step 1

First, let's find $\left(f \circ f^{-1}\right)(x)$: $\left(f \circ f^{-1}\right)(x) = f\left(f^{-1}(x)\right) = f\left(-\frac{2}{x}\right)$ Now, we plug in the expression for $f(x)$: $f\left(-\frac{2}{x}\right) = Show more…

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

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Key Concepts

Inverse Functions

Inverse functions are pairs of functions where one function undoes the effect of the other. If f and f?¹ are inverses, then applying f to f?¹(x) or f?¹ to f(x) returns the original input. This reversal property is essential in solving equations and understanding the relationship between functions.

Function Composition

Function composition is the process of applying one function to the result of another. It is denoted as (h ? g)(x) = h(g(x)). When working with inverse functions, composition demonstrates how one function can cancel the effect of its inverse, leading to the identity function.

Identity Function

The identity function is a special function that always returns its input, represented as I(x) = x. When a function is composed with its inverse, the result is the identity function, confirming that each operation undoes the other and that the functions are true inverses.

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