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)=9x+5 and g(x)=(x-5)/9 a. f(g(x))= (Simplify your answer.)

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)=9x+5 and g(x)=(x-5)/9
a. f(g(x))= (Simplify your answer.)

Transcript text: Find $\mathrm{f}(\mathrm{g}(\mathrm{x}))$ and $\mathrm{g}(\mathrm{f}(\mathrm{x}))$ and determine whether the pair of functions f and g are inverses of each other. \[ f(x)=9 x+5 \text { and } g(x)=\frac{x-5}{9} \] a. $f(g(x))=$ $\square$ (Simplify your answer.)

Solution

Solution Steps

To determine if two functions $ f $ and $ g $ are inverses of each other, we need to check if $ f(g(x)) = x $ and $ g(f(x)) = x $. We will first compute $ f(g(x)) $ by substituting $ g(x) $ into $ f(x) $ and simplifying the expression.

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

We start by substituting $ g(x) $ into $ f(x) $. Given the functions: \[ f(x) = 9x + 5 \quad \text{and} \quad g(x) = \frac{x - 5}{9} \] we compute $ f(g(x)) $: \[ f(g(x)) = f\left(\frac{x - 5}{9}\right) = 9\left(\frac{x - 5}{9}\right) + 5 \] Simplifying this expression: \[ f(g(x)) = (x - 5) + 5 = x \]

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

Next, we compute $ g(f(x)) $ by substituting $ f(x) $ into $ g(x) $: \[ g(f(x)) = g(9x + 5) = \frac{(9x + 5) - 5}{9} \] Simplifying this expression: \[ g(f(x)) = \frac{9x}{9} = x \]

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

Since we have found that: \[ f(g(x)) = x \quad \text{and} \quad g(f(x)) = x \] this confirms that the functions $ f $ and $ g $ are indeed inverses of each other.