Problem 2 - College Algebra 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. $$f(x)=6 x \text { and } g(x)=\frac{x}{6}$$

Step-by-Step Solution

Answer

The composed functions \(f(g(x))\) and \(g(f(x))\) both yield \(x\), thus demonstrating that \(f\) and \(g\) are indeed inverses of each other.

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

To compute \(f(g(x))\), replace the \(x\) in function \(f\) with function \(g(x)\). This gives us \(f(g(x)) = f\left(\frac{x}{6}\right)\). Now, substitute \(f(x)\) in the equation with the given \(f(x)=6x\). So, \(f(g(x)) = 6 \left(\frac{x}{6}\right)\). After multiplication, we get \(f(g(x))=x\).

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

Proceeding similarly, to compute \(g(f(x))\), replace \(x\) in function \(g\) with function \(f(x)\). This gives \(g(f(x)) = g(6x)\). Substituting \(g(x)\) with the given \(g(x)=\frac{x}{6}\), gives us \(g(f(x)) = \frac{6x}{6}\). After simplification, this too gives \(g(f(x))=x\).

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

Since both \(f(g(x))\) and \(g(f(x))\) resulted in \(x\), we can conclude that \(f\) and \(g\) are indeed inverses of each other. Two functions are inverses of each other if the composition of both functions in any order results in \(x\).