Chapter 4.2, Problem 21E

To determine

To calculate: To write the linear function

Answer to Problem 21E

The linear function is $f\left(x\right)=-3x+4$

Explanation of Solution

Given: $f\left(2\right)=-2;f\left(1\right)=1$

Formula Used:

Equation of line in slope intercept form is given as:

  $y=mx+c$ , where $\begin{array}{l}m=slope\\ c=y-intercept\end{array}$

Also, equation of line passing through two points is given as

  $y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)$

Calculation:

Given that values of linear function as follows:

  $f\left(2\right)=-2;f\left(1\right)=1$

This means that at $x=2$ , $y=-2$ and $x=1$ , $y=1$

Equation of function is found by finding the equation of line passing through $\left(2,-2\right)$ and $\left(1,1\right)$

Thus, equation of line is calculated as follows:

  $y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)$

Substituting the values in above equation:

  $\begin{array}{l}y-1=\frac{-2-1}{2-1}\left(x-1\right)\\ y-1=\frac{-3}{1}\left(x-1\right)\\ y-1=-3x+3\\ y=-3x+4\end{array}$

Conclusion:

Hence, the linear function is $f\left(x\right)=-3x+4$