Chapter 2.1, Problem 73AYU

To determine

To find:The function g .

Answer to Problem 73AYU

The function $g\left(x\right)$ is $5-\frac{7}{2}x$

Explanation of Solution

Given information:

A function $f\left(x\right)$ is given as $f\left(x\right)=3x+1$ and the composite function $\left(f+g\right)\left(x\right)=6-\frac{1}{2}x$ .

Calculation:

For any two functions $f\left(x\right)$ and $g\left(x\right)$ , the composite function $\left(f+g\right)\left(x\right)$ can be obtained by

adding the function $f\left(x\right)$ to the function $g\left(x\right)$ .

For a composite function $f+g$ , the domain of the function is defined by

  $\left(\text{Domain}\text{of}f\right)\cap \left(\text{Domain}\text{of}g\right)$

.So, consider the given functions,

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

Now,

  $\begin{array}{c}\left(f+g\right)\left(x\right)=6-\frac{1}{2}x\\ f\left(x\right)+g\left(x\right)=6-\frac{1}{2}x\\ 3x+1+g\left(x\right)=6-\frac{1}{2}x\\ g\left(x\right)=6-\frac{1}{2}x-3x-1\\ g\left(x\right)=5-\frac{7}{2}x\end{array}$

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