Chapter 4.1, Problem 22E

To determine

To writelinear function $f$ with the given values −

  $f\left(5\right)=-1$ and $f\left(0\right)=-5$

Answer to Problem 22E

The linear function is $f\left(x\right)=\frac{4}{5}x-5$ .

Explanation of Solution

Given:

  $f\left(5\right)=-1$

  $f\left(0\right)=-5$

Formula Used:

The equation of a line is written as $f\left(x\right)=mx+b$ where m is the slope and $b$ is the $y$ -intercept.

  $m=\frac{y_2-y_1}{x_2-x_1}$

Calculation:

Given:

  $f\left(5\right)=-1$

  $f\left(0\right)=-5$

As per the above definition, we have equation of the line as

  $f\left(x\right)=mx+b$

Since, $f\left(0\right)=-5$ , thus we have:

  $\begin{array}{l}-5=\left(m\times 0\right)+b\\ b=-5\end{array}$

Since, $f\left(5\right)=-1$ and $b=-5$ , thus we have:

  $\begin{array}{l}-1=\left(m\times 5\right)+\left(-5\right)\\ -1=5m-5\\ m=\frac{4}{5}\end{array}$

Thus, the equation of the line −

  $\begin{array}{l}f\left(x\right)=\left(\frac{4}{5}\times x\right)+\left(-5\right)\\ f\left(x\right)=\frac{4}{5}x-5\end{array}$

Thus, the linear function is $f\left(x\right)=\frac{4}{5}x-5$ .

Conclusion:

Hence, the linear function is $f\left(x\right)=\frac{4}{5}x-5$ .