Q. 56 - Precalculus Enhanced with Graphing Utilities

Question

Minimum Payments for Credit Cards Holders of credit cards issued by banks, department stores, oil companies, and so on, receive bills each month that state minimum amounts that must be paid by a certain due date. The minimum due depends on the total amount owed. One such credit card company uses the following rules: For a bill of less than $\backslash (10$, the entire amount is due. For a bill of at least $\backslash )10$ but less than $\backslash (500$, the minimum due is $\backslash )10$. A minimum of $\backslash (30$ is due on a bill of at least $\backslash )500$ but less than $\backslash (1000$, a minimum of $\backslash )50$ is due on a bill of at least $\backslash (1000$ but less than $\backslash )1500$, and a minimum of $\backslash (70$ is due on bills of $\backslash )1500$ or more. Find the function $f$that describes the minimum payment due on a bill of $x$ dollars. Graph $f$.

Step-by-Step Solution

Answer

The function is 

 $$\begin{gathered} f\left(x\right)=\{x, \text{ if }x<10 \\ 10, \text{ if }10\le x<500 \\ 30, \text{ if }500\le x<1000 \\ 50, \text{ if }1000\le x<1500 \\ 70, \text{ if }x\ge 1500 \end{gathered}$$

Step 1: Given information

Given the minimum amount is $\$10$

Step 2: Calculating the function

$\$500$Here for a bill of less than $\$10$, the entire amount is due, sox, if $x<10$

and for a bill of at least $\$10$ but less than , the minimum due is $\$10\Rightarrow 10$, if $10\le x<500$.

For a bill of at least $\$500$ but less than $\$1000$, the minimum due is $\$30\Rightarrow 30$, if $500\le x<1000$.

For a bill of at least $\$1000$ but less than $\$1500$, the minimum due is $\$50\Rightarrow 50$, if $1000\le x<1500$.

And for a bills of $\$ 1500$ or more, the minimum due is $\$70\Rightarrow 70$, if $x\ge 1500$.

Step 3: Forming the function

The function is

 $$\begin{gathered} f\left(x\right)=\{x,x<10 \\ 10, \text{ if }10\le x<500 \\ 30, \text{ if }500\le x<1000 \\ 50, \text{ if }1000\le x<1500 \\ 70, \text{ if }x\ge 1500 \end{gathered}$$