Question

In Exercises \(71-74,\) use the functions \(f(x)=x+4\) and \(g(x)=2 x-5\) to find the indicated function. $$ g^{-1} \circ f^{-1} $$

Short answer

The solution to the exercise is \(g^{-1} \circ f^{-1} = (x+1)/2\)

Step-by-step solution

Find inverse of \(f(x)=x+4\)

The inverse of a function \(f(x)\) is such that when applied to \(f(x)\), it returns the original input. It effectively 'undoes' the action of \(f(x)\). So, to find the inverse of \(f(x)=x+4\), we first replace \(f(x)\) with \(y\), thus getting \(y = x + 4\). Swap \(x\) and \(y\) to get \(x = y + 4\). Then solve for \(y\) to get: \(y = x - 4\). This is the inverse function, represented as \(f^{-1}(x) = x - 4\)

Find inverse of \(g(x)=2x-5\)

Apply similar steps to find the inverse of the function \(g(x) = 2x - 5\). Replace \(g(x)\) with \(y\) to get \(y = 2x -5\). Swap \(x\) and \(y\) to get \(x = 2y -5\). Solve for \(y\) to get \(y = (x + 5)/2\). This is the inverse function, represented as \(g^{-1}(x) = (x+5)/2\)

Find the composition \(g^{-1} \circ f^{-1}\)

We are asked to find \(g^{-1} \circ f^{-1}\), which means we apply \(g^{-1}\) to \(f^{-1}(x)\), or in other words substitute \(f^{-1}(x)\) into \(g^{-1}(x)\). This gives us \(g^{-1}(f^{-1}(x)) = g^{-1}(x-4) = \((x-4 +5)/2 = (x+1)/2\)