Problem 112 - 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. $$f^{-1} \circ g^{-1}$$

Step-by-Step Solution

Answer

The function \(f^{-1} \circ g^{-1}\) equals \(x/2 - 2\).

Step 1: Find the inverse of function \(f(x)\)

The given \(f(x) = x+4\). To find the inverse, let \(y = f(x)\), which gives \(y = x + 4\). By rearranging this equation to solve for \(x\), we get \(x = y - 4\). Substituting \(f^{-1}(y)\) for \(x\), we get \(f^{-1}(y) = y - 4\).

Step 2: Find the inverse of function \(g(x)\)

The given \(g(x) = 2x - 5\). To find the inverse, let \(y = g(x)\), which gives \(y = 2x - 5\). By rearranging this equation to solve for \(x\), we get \(x = (y + 5)/2\). Substituting \(g^{-1}(y)\) for \(x\), we get \(g^{-1}(y) = (y + 5)/2\).

Step 3: Find the composition \(f^{-1} \circ g^{-1}\)

The notation \(f^{-1} \circ g^{-1}\) means that the function \(g^{-1}\) is evaluated first, and then the function \(f^{-1}\) is evaluated. That is, \(f^{-1}(g^{-1}(x)) = f^{-1}((x+5)/2) = (x+5)/2 - 4 = x/2 - 2\). So, the function \(f^{-1} \circ g^{-1}\) equals \(x/2 - 2\).