Problem 114 - 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. $$(g \circ f)^{-1}$$

Step-by-Step Solution

Answer

The inverse of the composite function (g o f)^-1 is given by y = (x - 3)/2

Step 1: Calculate the composite function g(f(x))

Firstly, substitute f(x) into g(x). Given by \(f(x) = x + 4\) and \(g(x) = 2x - 5\), we substitute \(f(x)\) in for \(x\) in \(g(x)\) to get \(g(f(x)) = 2(f(x)) - 5 = 2(x + 4) - 5 = 2x + 8 - 5 = 2x + 3\)

Step 2: Find the inverse of the composite function

To find the inverse of a function y = g(f(x)), denoted by (g(f(x)))^-1, we swap x's and y's and solve for y. Let y = 2x + 3, exchanging x and y gives us x = 2y + 3. Solving this equation for \(y\), we get \(y = (x - 3)/2.\) This is the inverse of the composite function.