Problem 95 - College Algebra Essentials

Question

If \(f(x)=3 x\) and \(g(x)=x+5,\) find \((f \circ g)^{-1}(x)\) and \(\left(g^{-1} \circ f^{-1}\right)(x)\)

Step-by-Step Solution

Answer

So \((f \circ g)^{-1}(x) = (x - 15)/3\) and \((g^{-1} \circ f^{-1}\right)(x) = (x/3) - 5\).

Step 1: Find the composition of f and g

The composition function \(f \circ g(x)\) can be determined by substituting the g(x) into f(x). So here, \(f(g(x)) = f(x+5) = 3*(x+5) = 3x + 15\).

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

To find the inverse of the composed function, you must first replace the function with y: \(y = 3x + 15\). Solving this equation for x, you get \(x = (y - 15)/3\). Swap y and x to give the inverse function \((f \circ g)^{-1}(x) = (x - 15)/3\).

Step 3: Find inverse of f and g separately

The inverse of the function f(x) = 3x is \(f^{-1}(x) = x/3\), and the inverse of the function g(x) = x + 5 is \(g^{-1}(x) = x - 5\).

Step 4: Find the composition of \(g^{-1} \circ f^{-1}\)

The composition function \(g^{-1} \circ f^{-1}(x)\) can be determined by substituting the \(f^{-1}(x)\) into \(g^{-1}(x)\). So here, \(g^{-1}(f^{-1}(x))) = g^{-1}(x/3) = (x/3) - 5\).