Chapter 1.6, Problem 14E

a.

To determine

To write the linear function $f$ that has given function values $f\left(\frac{3}{5}\right)=\frac{1}{2},f(4)=5$

Answer to Problem 14E

The required function is $f(x)=\left(\frac{45}{34}\right)\left(x-4\right)+5$

Explanation of Solution

Given information :

Given function values $f\left(\frac{3}{5}\right)=\frac{1}{2},f(4)=5$ .

As we have $\left(x_1,y_1\right)=\left(\frac{3}{5},\frac{1}{2}\right),\left(x_2,y_2\right)=\left(4,5\right)$

Slope $m=\frac{y_2-y_1}{x_2-x_1}$

  $\begin{array}{l}\Rightarrow m=\frac{y_2-y_1}{x_2-x_1}\\ =\frac{5-\frac{1}{2}}{4-\frac{3}{5}}\\ =\frac{\frac{10-1}{2}}{\frac{20-3}{5}}\\ =\frac{9/2}{17/5}\end{array}$

  $\begin{array}{l}\Rightarrow m=\frac{9}{2}\times \frac{5}{17}\\ \therefore m=\frac{45}{34}\end{array}$

Thus, the required function is that has slope $m=\frac{45}{34}$ and passes through the point $\left(x_2,y_2\right)=\left(4,5\right)$ is $y-5=\left(\frac{45}{34}\right)\left(x-4\right)$

  $\Rightarrow y=\left(\frac{45}{34}\right)\left(x-4\right)+5$

The required function is $f(x)=\left(\frac{45}{34}\right)\left(x-4\right)+5$

b.

To determine

To sketch the graph of the function $f(x)=\left(\frac{45}{34}\right)\left(x-4\right)+5$

Explanation of Solution

Given information :

Given function $f(x)=\left(\frac{45}{34}\right)\left(x-4\right)+5$

For $x=0$ then $y-$ intercept is $f(x)=y=-0.2941$

For $y=0$ then $x-$ intercept is $\Rightarrow x=0.222$

Graph: The graph of the function is $f(x)=\left(\frac{45}{34}\right)\left(x-4\right)+5$ is

  

Interpretation: From above graph, we can observe that the given function has $x\text{and}y-$ intercepts as $0.2222\text{and}-0.2941$ respectively .