Chapter 2.1, Problem 73AYU

Given $f\left(x\right)\text{ = 3}x\text{ + 1}$ and $\left(f\text{ + }g\right)\left(x\right)\text{ = 6 - }\frac{1}{2}x$, find the function $g$.

To determine

To find: The function $g$ if $f\left(x\right)=3x+1$ and $\left(f+g\right)\left(x\right)=6-\frac{1}{2}x$

Answer to Problem 73AYU

$g\left(x\right)=5-\frac{7}{2}x$

Explanation of Solution

Given:

The function $f\left(x\right)=3x+1$; $\left(f+g\right)\left(x\right)=6-\frac{1}{2}x$

Calculation:

$\left(f+g\right)\left(x\right)=f\left(x\right)+g\left(x\right)$

$6-\frac{1}{2}x=3x+1+g\left(x\right)$

$6-\frac{1}{2}x-3x-1=g\left(x\right)$

$\text{5 }-\left(\frac{1}{2}x+3x\right)=g\left(x\right)$

$g\left(x\right)=5-\frac{7}{2}x$