Chapter 4.2, Problem 22E

To determine

To calculate: To write the linear function

Answer to Problem 22E

The linear function is $f\left(x\right)=x+2$

Explanation of Solution

Given: $f\left(5\right)=7;f\left(-2\right)=0$

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:

  $f\left(5\right)=7;f\left(-2\right)=0$

This means that at $x=5$ , $y=7$ and $x=-2$ , $y=0$

Equation of function is found by finding the equation of line passing through $\left(5,7\right)$ and $\left(-2,0\right)$

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:

  $\begin{array}{l}y-0=\frac{0-7}{-2-5}\left(x-\left(-2\right)\right)\\ y=\frac{-7}{-7}\left(x+2\right)\\ y=1\left(x+2\right)\\ y=x+2\end{array}$

Conclusion:

Hence, the linear function is $f\left(x\right)=x+2$