Questions: Let f(x) = 1/(x-6) and g(x) = 3/x + 6. Find the following functions. Simplify your answers. f(g(x))= g(f(x))=

Transcript text: Let $f(x)=\frac{1}{x-6}$ and $g(x)=\frac{3}{x}+6$. Find the following functions. Simplify your answers. \[ f(g(x))= \] $\square$ \[ g(f(x))= \] $\square$ Question Help: Video Submit Question

Solution

Solution Steps

To find the composite functions $ f(g(x)) $ and $ g(f(x)) $, we need to substitute $ g(x) $ into $ f(x) $ and vice versa. Simplify the resulting expressions to get the final answers.

Step 1: Define the Functions

We start with the given functions: \[ f(x) = \frac{1}{x - 6} \] \[ g(x) = \frac{3}{x} + 6 \]

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

To find $ f(g(x)) $, we substitute $ g(x) $ into $ f(x) $: \[ f(g(x)) = f\left(\frac{3}{x} + 6\right) = \frac{1}{\left(\frac{3}{x} + 6\right) - 6} \] Simplify the expression inside the denominator: \[ f(g(x)) = \frac{1}{\frac{3}{x}} = \frac{x}{3} \]

Step 3: Compute $ g(f(x)) $

To find $ g(f(x)) $, we substitute $ f(x) $ into $ g(x) $: \[ g(f(x)) = g\left(\frac{1}{x - 6}\right) = \frac{3}{\frac{1}{x - 6}} + 6 \] Simplify the expression: \[ g(f(x)) = 3(x - 6) + 6 = 3x - 18 + 6 = 3x - 12 \]