Chapter 1.5, Problem 96E

(a)

To determine

To match: The function $f\left(p\right):\text{A}50\%\text{discount is applied}$ with the graph.

  

Answer to Problem 96E

The graph of the function $f\left(p\right):\text{A}50\%\text{discount is applied}$ is $L_2$ .

Explanation of Solution

Given information:

The graphs labeled $L_1$ , $L_2$ , $L_3$ and $L_4$ represents four different pricing discounts, where $p$ is the original price in dollars and $S$ is the sale price(in dollars).

  

As the function $f\left(p\right):\text{A}50\%\text{discount is applied}$ says that the sale price becomes half the original price.

So, the function $f\left(p\right)=\frac{p}{2}$ represents the sale price for the function.

Therefore, the graph of the function $f\left(p\right):\text{A}50\%\text{discount is applied}$ is $L_2$ .

(b)

To determine

To match: The function $g\left(p\right):\text{A}\$5\text{discount is applied}$ with the graph.

  

Answer to Problem 96E

The graph of the function $g\left(p\right):\text{A}\$5\text{discount is applied}$ is $L_1$ .

Explanation of Solution

Given information:

The graphs labeled $L_1$ , $L_2$ , $L_3$ and $L_4$ represents four different pricing discounts, where $p$ is the original price in dollars and $S$ is the sale price(in dollars).

  

As the function $g\left(p\right):\text{A}\$5\text{discount is applied}$ says that the sale price becomes $\$5$ less than the original price.

So, the function $g\left(p\right)=p-5$ represents the sale price for the function.

Therefore, the graph of the function $g\left(p\right):\text{A}\$5\text{discount is applied}$ is $L_1$ .

(c)

To determine

To match: The function $\left(gof\right)\left(p\right)$ with the graph and also the condition.

  

Answer to Problem 96E

The graph of the function $\left(gof\right)\left(p\right)$ is $L_4$ and the condition for $\left(gof\right)\left(p\right)$ says that firstly $50\%$ discount is applied and then $\$5$ discount is applied.

Explanation of Solution

Given information:

The graphs labeled $L_1$ , $L_2$ , $L_3$ and $L_4$ represents four different pricing discounts, where $p$ is the original price in dollars and $S$ is the sale price(in dollars).

  

As calculated in part(a), the function $f\left(p\right)=\frac{p}{2}$ and as calculated in part(b), the function $g\left(p\right)=p-5$ .

Simplify the composition.

  $\begin{array}{c}\left(gof\right)\left(p\right)=g\left(f\left(p\right)\right)\\ =g\left(\frac{p}{2}\right)\\ =\frac{p}{2}-5\end{array}$

The condition for $\left(gof\right)\left(p\right)$ says that firstly $50\%$ discount is applied and then $\$5$ discount is applied.

Therefore, the graph of the function $\left(gof\right)\left(p\right)$ is $L_4$ and the condition for $\left(gof\right)\left(p\right)$ says that firstly $50\%$ discount is applied and then $\$5$ discount is applied.

(d)

To determine

To match: The function $\left(fog\right)\left(p\right)$ with the graph.

  

Answer to Problem 96E

The graph of the function $\left(fog\right)\left(p\right)$ is $L_3$ and the condition for $\left(fog\right)\left(p\right)$ says that firstly $\$5$ discount is applied and then $50\%$ discount is applied.

Explanation of Solution

Given information:

The graphs labeled $L_1$ , $L_2$ , $L_3$ and $L_4$ represents four different pricing discounts, where $p$ is the original price in dollars and $S$ is the sale price(in dollars).

  

As calculated in part(a), the function $f\left(p\right)=\frac{p}{2}$ and as calculated in part(b), the function $g\left(p\right)=p-5$ .

Simplify the composition.

  $\begin{array}{c}\left(fog\right)\left(p\right)=f\left(g\left(p\right)\right)\\ =f\left(p-5\right)\\ =\frac{p-5}{2}\end{array}$

The condition for $\left(fog\right)\left(p\right)$ says that firstly $\$5$ discount is applied and then $50\%$ discount is applied.

Therefore, the graph of the function $\left(fog\right)\left(p\right)$ is $L_3$ and the condition for $\left(fog\right)\left(p\right)$ says that firstly $\$5$ discount is applied and then $50\%$ discount is applied.