Chapter 2.4, Problem 56AYU

To determine

The function, and graph the function that describe the minimum payment due on a bill of $x$ dollars. Where, the rules are;

For a bill of less than $\$10$ the entire amount is due.

For a bill at least $\$10$ but less than $\$500$, the minimum due is $\$10$ .

A minimum of $\$30$ is due on a bill of at least $\$500$ but less than $\$1000$ .

A minimum of $\$50$ is due on a bill of at least $\$1000$ but less than $\$1500$ .

A minimum of $\$70$ is due on bills of $\$1500$ or more.

Answer to Problem 56AYU

Solution:

The function $y=f(x)$ describes the minimum payment due on a bill of $x$ dollars is

  $y=f(x)=\{\begin{array}{l}x\text{if}0\le x<10\\ 10\text{if}10\le x<500\\ 30\text{if}500\le x<1000\\ 50\text{if}1000\le x<1500\\ 70\text{if}1500\le x\end{array}$ .

The graph of function is

  

Explanation of Solution

Given information:

One such credit card company uses the following rules:

For a bill of less than $\$10$ the entire amount is due.

For a bill at least $\$10$ but less than $\$500$, the minimum due is $\$10$ .

A minimum of $\$30$ is due on a bill of at least $\$500$ but less than $\$1000$ .

A minimum of $\$50$ is due on a bill of at least $\$1000$ but less than $\$1500$ .

A minimum of $\$70$ is due on bills of $\$1500$ or more.

The function $y=f(x)$, describe the minimum payment due on a bill of $x$ dollars is as below:

From the given information,

For a bill of less than $\$10$, the minimum payment is the entire amount $\$x$

That is, the minimum payment due $f(x)$ for a bill $0\le x\le 10$ is $f(x)=x$ … (1)

For a bill at least $\$10$ but less than $\$500$, the minimum due is $\$10$ .

That is, the minimum payment due $f(x)$ for a bill $10\le x<500$ is $f(x)=10$ … (2)

For a bill of at least $\$500$ but less than $\$1000$, a minimum due is $\$30$ .

That is, the minimum payment due $f(x)$ for a bill $500\le x<1000$ is $f(x)=30$ … (3)

For a bill of at least $\$1000$ but less than $\$1500$, a minimum due is $\$50$ .

That is, the minimum payment due $f(x)$ for a bill $1000\le x<1500$ is $f(x)=50$ … (4)

For a bills of $\$1500$ or more, a minimum due is $\$70$ .

That is, the minimum payment due $f(x)$ for a bill $x\ge 1500$ is $f(x)=70$ … (4)

From (1) to (5),

Therefore, the function $y=f(x)$ describe the minimum payment due on a bill of $x$ dollars is $y=f(x)=\{\begin{array}{l}x\text{if}0\le x<10\\ 10\text{if}10\le x<500\\ 30\text{if}500\le x<1000\\ 50\text{if}1000\le x<1500\\ 70\text{if}1500\le x\end{array}$ .

The graph of function $y=f(x)$, describes the minimum payment due on a bill of $x$ dollars is as below: