Chapter 3.5, Problem 51PPS

a.

To determine

To write: A function $f\left(x\right)$ for the given situation.

Answer to Problem 51PPS

  $f\left(x\right)=70x+25$

Explanation of Solution

Given:

The initial subscription fee is $25, and the two-meal plan costs $70 per week.

Calculation:

Two-meal costs for $x$ weeks is $\$70x$ .

So, the function $f\left(x\right)$ can be written as:

  $\begin{array}{c}f\left(x\right)=\text{variable cost }+\text{ fixed cost}\\ =70x+25\end{array}$

b.

To determine

To write: A function $h\left(x\right)$ for the given situation.

Answer to Problem 51PPS

  $h\left(x\right)=70x+30$

Explanation of Solution

Given:

The initial subscription fees changed to $30, and the two-meal plan costs $70 per week.

Calculation:

Two-meal costs for $x$ weeks is $\$70x$ .

So, the function $f\left(x\right)$ can be written as:

  $\begin{array}{c}h\left(x\right)=\text{variable cost }+\text{ fixed cost}\\ =70x+30\end{array}$

c.

To determine

To describe: The transformation from $f\left(x\right)$ to $h\left(x\right)$ .

Answer to Problem 51PPS

  $f\left(x\right)$ is shifted 5 units upward to get $h\left(x\right)$ .

Explanation of Solution

Observe that the $h\left(x\right)$ is obtained by adding 5 to $f\left(x\right)$ . That is,

  $h\left(x\right)=f\left(x\right)+5$ .

That means, the graph of $f\left(x\right)$ is shifted 5 units upward to get the graph of $h\left(x\right)$ .

d.

To determine

To write: The function $p\left(x\right)$ .

Answer to Problem 51PPS

  $p\left(x\right)=63x+30$

Explanation of Solution

Given:

A promotion is offered for 10% off if ordered online.

Calculation:

There will not be any change in the subscription price. After 10%, the costs of two-meal plan for x weeks will be:

  $\begin{array}{c}p\left(x\right)=\left(70x-10\%\text{of }70x\right)+30\\ =70x-7x+30\\ =63x+30\end{array}$

e.

To determine

To describe: The transformation from $h\left(x\right)$ to $p\left(x\right)$ .

Answer to Problem 51PPS

  $h\left(x\right)$ is stretched in $x$ -direction to get the graph of $p\left(x\right)$ .

Explanation of Solution

The function $h\left(x\right)$ is $h\left(x\right)=70x+30$ .

The function $p\left(x\right)$ can be written as $p\left(x\right)=\frac{9}{10}\left(70x\right)+30$ .

Thus, the function $h\left(x\right)$ is stretched in $x$ -direction to get the graph of $p\left(x\right)$ .