Chapter 3.9, Problem 18WE

To determine

To find: An equation of the linear function $f$ .

Answer to Problem 18WE

Therefore, the equation of linear function $f\left(x\right)=-3x+12$ .

Explanation of Solution

Given information: The function $f\text{ at }x=2$ and $f\text{ at }x=4$ ,

  $f\left(2\right)=6,f\left(4\right)=0$

Formula used: For calculation of equation of linear function,

  $\begin{array}{l}f\left(x\right)=mx+b\\ \text{where }m\text{ is slope and b is }y-\text{intercept}\text{.}\end{array}$

Calculation: To calculate the equation of function using function $f\text{ at }x=2\text{ and }f\text{at }x=4$ ,

  $\begin{array}{l}\text{Putting value of }f\left(2\right)\text{in genral form,}\\ f\left(x\right)=mx+b\\ \text{Now, at }x=2\\ f\left(2\right)=m\times 2+b\\ 6=2m+b\\ b=6-2m\\ \text{Putting value of }f\left(4\right)\text{in genral form,}\\ f\left(x\right)=mx+b\\ \text{Now, at }x=4\\ f\left(4\right)=m\times 4+b\\ 0=4m+b\\ \text{Putting value of }b\\ 0=4m+(6-2m)\\ m=-3\\ \text{Now,}b=6-2m\\ b=12\\ f\left(x\right)=-3x+12\end{array}$

Therefore, the equation of linear function $f\left(x\right)=-3x+12$ .