Problem 111 - 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. $$g^{-1} \circ f^{-1}$$

Step-by-Step Solution

Answer

The composition of the inverse functions \(g^{-1} \circ f^{-1}(x)\) is \((x+1)/2\).

Step 1: Find the Inverse of Function f

To find the inverse of function f, denoted by \(f^{-1}(x)\), we interchange \(f(x)\) with \(x\) and solve for \(x\). In this case, \(f(x) = x + 4\) becomes \(f^{-1}(x) = x - 4\).

Step 2: Find the Inverse of Function g

To obtain \(g^{-1}(x)\), we repeat a similar process. We interchange \(g(x)\) with \(x\) and solve for \(x\). Given \(g(x) = 2x - 5\), the inverse function is determined as follows: \(g^{-1}(x) = (x + 5)/2\).

Step 3: Determine the Composition

Now, to obtain \(g^{-1} \circ f^{-1}(x)\), we have to substitute \(f^{-1}(x)\) into \(g^{-1}(x)\). Plugging \(x - 4\) from \(f^{-1}(x)\) into \(g^{-1}(x) = (x + 5)/2\), we get \(g^{-1} \circ f^{-1}(x) = ((x - 4) + 5)/2 = (x + 1)/2\).