Problem 5 - Precalculus Essentials

Question

Find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. $$f(x)=5 x-9 \text { and } g(x)=\frac{x+5}{9}$$

Step-by-Step Solution

Answer

Yes, the functions \(f(x) = 5x - 9\) and \(g(x) = \frac{x+5}{9}\) are inverses of each other as the compositions \(f(g(x))\) and \(g(f(x))\) both return \(x\).

Step 1: Calculate \(f(g(x))\)

Begin by plugging the function \(g(x) = \frac{x+5}{9}\) into the function \(f(x) = 5x - 9\). It would look like this: \(f(g(x)) = 5(\frac{x+5}{9}) - 9\). Simplify this to get \(f(g(x)) = x\).

Step 2: Calculate \(g(f(x))\)

Now, you need to plug the function \(f(x) = 5x - 9\) into the function \(g(x) = \frac{x+5}{9}\). It would look like this: \(g(f(x)) = \frac{(5x - 9) + 5}{9}\). Simplify this to get \(g(f(x)) = x\).

Step 3: Conclude if \(f\) and \(g\) are inverses

Since the results from Step 1 and Step 2 are both \(x\), it can be concluded that \(f\) and \(g\) are indeed inverses of each other because \(f(g(x)) = x\) and \(g(f(x)) = x\). This implies that for every \(x\) in the domain, the compositions of \(f\)and \(g\) still return \(x\) as the result.