Q. 73 - Precalculus Enhanced with Graphing Utilities

Question

Given $f\left(x\right)=3x+1$ and $(f+g)\left(x\right)=6-12x$, find the function $g$.

Step-by-Step Solution

Answer

The value of $g\left(x\right)=5(1-3x)$

Step 1. Given Information

Given$f\left(x\right)=3x+1$and$(f+g)\left(x\right)=6-12x$

We have to find the function$g$.

Step 2. We know that ( f + g ) ( x ) = f ( x ) + g ( x ) Simplifying the formula for g ( x )

 Subtract $f\left(x\right)$ on both side

$$\begin{gathered} (f+g)\left(x\right)-f\left(x\right)=f\left(x\right)+g\left(x\right)-f\left(x\right) \\ (f+g)\left(x\right)-f\left(x\right)=g\left(x\right) \\ g\left(x\right)=(f+g)\left(x\right)-f\left(x\right) \end{gathered}$$

Step 3. Putting the value of f ( x )   and   ( f + g ) ( x )

$$\begin{gathered} g\left(x\right)=6-12x-\left(3x+1\right) \\ g\left(x\right)=6-12x-3x-1 \end{gathered}$$

Simplify

$$\begin{gathered} g\left(x\right)=6-12x-3x-1 \\ g\left(x\right)=5-15x \\ g\left(x\right)=5(1-3x) \end{gathered}$$