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Chapter 1: Problem 92

Use the functions \(f(x)=x+4\) and \(g(x)=2 x-5\) to find the specified function. $$(g \circ f)^{-1}$$

Short Answer

Expert verified
The inverse of the composite function \(g(f(x))\) is \((g \circ f)^{-1}(x)=(x-3)/2\).

Step by step solution

01

Composition of Functions

The first step is to find the composite function \(g(f(x))\), which means replacing every \(x\) in \(g(x)\) with the function \(f(x)\).\nHence, we get \(g(f(x))=2(f(x))-5=2(x+4)-5=2x+8-5=2x+3.\)
02

Inverse of the Composite Function

The second step is to find the inverse of the resulting function. To obtain the inverse \((g \circ f)^{-1}\), replace the \(y\) in the equation with \(x\) and solve for \(y\).\nHence, \(x=2y+3\) which can further be solved as follows to give \(y=(x-3)/2\). Hence, \((g \circ f)^{-1}(x)=(x-3)/2\).

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