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Chapter 2: Problem 70

Explain how to determine if two functions are inverses of each other.

Short Answer

Expert verified
Two functions \(f\) and \(g\) are inverses of each other if performing the function composition \(f(g(x))\) and \(g(f(x))\) results in \(x\), for all values of \(x\) in their domain.

Step by step solution

01

Understanding Inverse Functions

When we say that two functions are inverses of each other, it means that one function 'undoes' the operation of the other. Mathematically, if we have two functions \(f\) and \(g\), they are inverses of each other if for every input \(x\), \(f(g(x)) = g(f(x)) = x\). This condition should hold true for all values in the domain of \(x\).
02

Function Composition

To check if the two functions are inverses, we need to perform function composition. Function composition involves applying one function to the results of another: we first apply function \(g\) to \(x\) (i.e., we get \(g(x)\)), and then we apply function \(f\) to \(g(x)\). The result should be equal to \(x\). Similarly, we need to perform the composition in the other direction: apply \(f\) to \(x\) (i.e., \(f(x)\)), then apply \(g\) to \(f(x)\). The result should again equal \(x\).
03

Verify The Condition

If the equations \(f(g(x)) = x\) and \(g(f(x)) = x\) hold true for all \(x\) in the domain of \(f\) and \(g\), then \(f\) and \(g\) are inverses of each other.

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Most popular questions from this chapter

Find a. \((f \circ g)(x)\) b. the domain of \(f \circ g\) $$f(x)=x^{2}+1, g(x)=\sqrt{2-x}$$

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