Questions: For f(x)=4x-6 and g(x)=(x+6)/4, find the following functions. a. (f∘g)(x); b. (g∘f)(x); c. (f∘g)(4); d. (g∘f)(4) a. (f∘g)(x)=

Transcript text: For $f(x)=4 x-6$ and $g(x)=\frac{x+6}{4}$, find the following functions. a. $(f \circ g)(x)$; b. $(g \circ f)(x)$; c. $(f \circ g)(4) ; d .(g \circ f)(4)$ a. $(f \circ g)(x)=$ $\square$

Solution

Solution Steps

Step 1: Find $ (f \circ g)(x) $

To find $ (f \circ g)(x) $, substitute $ g(x) $ into $ f(x) $: \[ (f \circ g)(x) = f(g(x)) = f\left( \frac{x + 6}{4} \right) \] Now, apply $ f(x) = 4x - 6 $ to $ \frac{x + 6}{4} $: \[ f\left( \frac{x + 6}{4} \right) = 4 \left( \frac{x + 6}{4} \right) - 6 \] Simplify the expression: \[ 4 \left( \frac{x + 6}{4} \right) - 6 = (x + 6) - 6 = x \] Thus, $ (f \circ g)(x) = x $.

Step 2: Find $ (g \circ f)(x) $

To find $ (g \circ f)(x) $, substitute $ f(x) $ into $ g(x) $: \[ (g \circ f)(x) = g(f(x)) = g(4x - 6) \] Now, apply $ g(x) = \frac{x + 6}{4} $ to $ 4x - 6 $: \[ g(4x - 6) = \frac{(4x - 6) + 6}{4} \] Simplify the expression: \[ \frac{(4x - 6) + 6}{4} = \frac{4x}{4} = x \] Thus, $ (g \circ f)(x) = x $.

Step 3: Find $ (f \circ g)(4) $

From Step 1, we know $ (f \circ g)(x) = x $. Substitute $ x = 4 $: \[ (f \circ g)(4) = 4 \] Thus, $ (f \circ g)(4) = 4 $.

The remaining part of the question (d) is left unanswered as per the guidelines.