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Chapter 5: Problem 82

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

Short Answer

Expert verified
\((g \circ f)^{-1}(x) = (x - 3)/2\

Step by step solution

01

Compute the Composite Function \(g \circ f\)

The first thing to do is to find the composition of functions \(g \circ f\). This means substituting function \(f(x)\) into function \(g(x)\). \(g(f(x)) = g(x+4) = 2(x+4) - 5 = 2x +8 - 5 = 2x + 3\). So, \(g \circ f = 2x + 3\)
02

Finding the Inverse of the Composite Function

After getting \(g \circ f = 2x + 3\), we need to find the inverse of this composite function. Set \(y = 2x + 3\). To find the inverse, we solve for \(x\) in terms of \(y\). This gives us \(x = (y - 3)/2\). Therefore, \((g \circ f)^{-1} = (y - 3)/2\)
03

Replace \(y\) With \(x\)

In practice, the variable used in function definition does not impact the nature of the function. Thus, we replace \(y\) with \(x\) to keep consistant with the variable used in the original function definitions. So \((g \circ f)^{-1}\) can be written as \((g \circ f)^{-1}(x) = (x - 3)/2\).

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