Chapter 4.2, Problem 31E

To determine

To calculate: To describe and correct the error in writing a linear function $g$ with the values $g\left(5\right)=4$ and $g\left(3\right)=10$

Answer to Problem 31E

The linear function is $g\left(x\right)=-3x+19$

Explanation of Solution

Given: Step in writing the linear function $g$ is

  $\begin{array}{l}m=\frac{10-4}{3-5}\\ m=\frac{6}{-2}=-3\end{array}$$\begin{array}{l}y-y_1=mx-x_1\\ y-4=-3x-5\\ y=-3x-1\end{array}$

A function is $g\left(x\right)=-3x-1$

Formula Used:

If the slope of line is constant, then the data can be modelled by a linear equation.

Also, equation of line passing through point and given slope is

  $y-y_1=m\left(x-x_1\right)$

Calculation:

Values of function is given as follows:

  $g\left(5\right)=4$ and $g\left(3\right)=10$

Thus, slope is calculated as

  $\begin{array}{l}m=\frac{10-4}{3-5}\\ m=\frac{6}{-2}=-3\end{array}$

Equation of linear function is calculated as follows:

  $\begin{array}{l}y-y_1=m\left(x-x_1\right)\\ y-y_1=mx-m{x}_1\end{array}$

Substituting the values

  $\begin{array}{l}y-4=-3x-\left(-3\right)5\\ y-4=-3x+15\\ y=-3x+19\end{array}$

Conclusion:

Hence, the linear function is $g\left(x\right)=-3x+19$