Chapter 4.1, Problem 26E

To determine

To writelinear function $f$ with the given values −

  

Answer to Problem 26E

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

Explanation of Solution

Given:

  $f\left(-4\right)=-2$

  $f\left(-2\right)=-1$

  $f\left(0\right)=0$

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:

  $f\left(-4\right)=-2$

  $f\left(-2\right)=-1$

  $f\left(0\right)=0$

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

  $f\left(x\right)=mx+b$

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

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

Since, $f\left(-4\right)=-2$ and $b=0$ , thus we have:

  $\begin{array}{l}-2=\left(m\times -4\right)+0\\ m=\frac{1}{2}\end{array}$

Thus, the equation of the line −

  $\begin{array}{l}f\left(x\right)=\left(\frac{1}{2}\times x\right)+0\\ f\left(x\right)=\frac{1}{2}x\end{array}$

Thus, the linear function is $f\left(x\right)=\frac{1}{2}x$ .

Conclusion:

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