Chapter 3.10, Problem 4ST

To determine

To calculate:The equation of the linear function f for the given information.

Answer to Problem 4ST

The linear function is $f\left(x\right)=-\frac{2}{3}x+1$ .

Explanation of Solution

Given information:The value of $f\left(x\right)\text{decreases}\text{by}4\text{when}x\text{increases}\text{by}\text{6}$ and $f\left(3\right)=-1$ .

Formula used:

The equation of a linear function is given by

  $f\left(x\right)=mx+b,\text{where}m\text{is}\text{the}\text{slope}\text{of}\text{the}\text{function}$ .

Calculation:

Let the linear function be $f\left(x\right)=mx+b$ ……(1)

When $f\left(x\right)\text{decreases}\text{by}4\text{when}x\text{increases}\text{by}\text{6}$

Therefore

  $\begin{array}{l}m=\frac{\text{change}\text{in}f\left(x\right)}{\text{change}\text{in}x}=\frac{\Delta y}{\Delta x}\\ m=\frac{-4}{6}=-\frac{2}{3}\end{array}$

Put $m=-\frac{2}{3}$ and $f\left(3\right)=-1$ in equation (1).

  $\begin{array}{l}f\left(3\right)=-\frac{2}{3}\left(3\right)+b\\ -1=-2+b\\ b=1\end{array}$

Then the linear function is

  $\begin{array}{l}f\left(x\right)=mx+b,\text{where}m=-\frac{2}{3}\&b=1\\ f\left(x\right)=-\frac{2}{3}x+1\end{array}$

Hence the linear function is $f\left(x\right)=-\frac{2}{3}x+1$ .