Questions: Find (f(g(x))) and (g(f(x))) and determine whether the pair of functions (f) and (g) are inverses of each other. (f(x)=9 x text and g(x)=fracx9) a. (f(g(x))=) (square)

$Find (f(g(x))) and (g(f(x))) and determine whether the pair of functions (f) and (g) are inverses of each other. (f(x)=9 x text and g(x)=fracx9) a. (f(g(x))=) (square)$

Transcript text: Find $f(g(x))$ and $g(f(x))$ and determine whether the pair of functions $f$ and $g$ are inverses of each other. \[ f(x)=9 x \text { and } g(x)=\frac{x}{9} \] a. $f(g(x))=$ $\square$

Solution

Solution Steps

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

We start by substituting $ g(x) $ into $ f(x) $: \[ f(g(x)) = f\left(\frac{x}{9}\right) = 9 \left(\frac{x}{9}\right) = x \]

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

Next, we substitute $ f(x) $ into $ g(x) $: \[ g(f(x)) = g(9x) = \frac{9x}{9} = x \]

Step 3: Determine if $ f $ and $ g $ are inverses

Since both compositions yield the identity function: \[ f(g(x)) = x \quad \text{and} \quad g(f(x)) = x \] we conclude that $ f $ and $ g $ are indeed inverses of each other.