Problem 113 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

The inverse of the function obtained by the composition of \(f(x)=x+4\) and \(g(x)=2x-5\) is \((f \circ g)^{-1}(x)=(x+1)/2\).

Step 1: Compose the Functions

Apply the functions sequentially as specified by the composition \(f \circ g\). That means to apply function \(g\) first, then function \(f\). In this case, substitute \(g(x)=2x-5\) into \(f(x)=x+4\). Thus, the composition \(f(g(x))=f(2x-5)=(2x-5) + 4 = 2x-1\).

Step 2: Compute the Inverse Function

To find the inverse of \(f(g(x))=2x-1\), we replace \(f(g(x))\) with \(y\) to get \(y=2x-1\). Interchange \(x\) and \(y\) to get \(x=2y-1\), then solve for \(y\) to obtain the inverse function: Adding 1 and dividing by 2 gives the inverse function \((f \circ g)^{-1}(x)=(x+1)/2\).

Step 3: Check the Inverse Function

To confirm that the function obtained is truly the inverse, we can check by verifying whether \((f \circ g) \circ (f \circ g)^{-1}(x)=x\) and \((f \circ g)^{-1} \circ (f \circ g)(x)=x\). Substituting \((f \circ g)^{-1}(x)=(x+1)/2\) into the equation results in \((f \circ g) \circ (f \circ g)^{-1}(x)=2((x+1)/2) -1 = x\), and \((f \circ g)^{-1} \circ (f \circ g)(x)=((2x-1)+1)/2=x\). Thus, the function obtained is indeed the inverse.