Problem 4 - College Algebra

Question

Find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. $$f(x)=4 x+9 \text { and } g(x)=\frac{x-9}{4}$$

Step-by-Step Solution

Answer

Yes, the functions \(f(x) = 4x + 9\) and \(g(x) = \frac{x - 9}{4}\) are inverses of each other because both \(f(g(x))\) and \(g(f(x))\) are equal to \(x\).

Step 1: Compose Function \(f(g(x))\)

First, let's compute \(f(g(x))\). This gets done by substituting \(g(x)\) into \(f(x)\). In this case that implies replacing all instances of \(x\) in the equation for \(f(x)\) with the function \(g(x)\). That gives: \(f(g(x)) = f(\frac{x-9}{4}) = 4(\frac{x-9}{4}) + 9 = x + 9 - 9 = x.\)

Step 2: Compose Function \(g(f(x))\)

Now perform the second composition, that is, compute \(g(f(x))\). As before, we substitute \(f(x)\) into \(g(x)\), replacing all instances of \(x\) in the equation for \(g(x)\) with the expression for \(f(x)\). Thus, \(g(f(x)) = g(4x + 9) = \frac{(4x + 9) - 9}{4} = \frac{4x}{4} = x.\)

Step 3: Verify if \(f\) and \(g\) are Inverse

Lastly, revisit the definition of inverse functions. Two functions \(f\) and \(g\) are inverses of each other if \(f(g(x)) = x\) and \(g(f(x)) = x\) for all \(x\) in their domains. Both compositions resulted in \(x\), so \(f\) and \(g\) are indeed inverse functions.