Question

given f(x) = 2x + 6 and (f + g)(x) = 8 - \frac{1}{4}x, find the function g.
g(x) =
(simplify your answer. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall function - addition formula

We know that \((f + g)(x)=f(x)+g(x)\). So, \(g(x)=(f + g)(x)-f(x)\).

Step2: Substitute the given functions

Given \(f(x)=2x + 6\) and \((f + g)(x)=8-\frac{1}{4}x\), then \(g(x)=(8-\frac{1}{4}x)-(2x + 6)\).

Step3: Simplify the expression

\[

$$\begin{align*} g(x)&=8-\frac{1}{4}x-2x - 6\\ &=(8 - 6)+(-\frac{1}{4}x-2x)\\ &=2+(-\frac{1}{4}x-\frac{8}{4}x)\\ &=2-\frac{9}{4}x \end{align*}$$

\]

Answer:

\(2-\frac{9}{4}x\)