91Ó°ÊÓ

Rewrite the function \(g\) in the form \(g(x)=\frac{a}{x-h}+k\). Graph the function. Describe the graph of \(g\) as a transformation of the graph of \(f(x)=\frac{a}{x}\). \(g(x)=\frac{2 x+3}{x}\)

Give an example of a rational equation that you would solve using cross multiplication and one that you would solve using the LCD. Explain your reasoning.

In Exercises 33-40, rewrite the function in the form \(g(x)=\frac{a}{x-h}+k\). Graph the function. Describe the graph of \(g\) as a transformation of the graph of \(f(x)=\frac{a}{x}\). $$ g(x)=\frac{2 x-4}{x-5} $$

Manufacturers often package products in a way that uses the least amount of material. One measure of the effi ciency of a package is the ratio of its surface area to its volume. The smaller the ratio, the more effi cient the packaging. a. Write an expression for the efficiency ratio \(\frac{S}{V}\) of a cylindrical package. b. Find the efficiency ratio for each cylindrical can listed in the table. $$ \begin{array}{|l|c|c|c|} \cline { 2 - 4 } \multicolumn{1}{c|}{} & \text { Soup } & \text { Coffee } & \text { Paint } \\ \hline \text { Height, } \boldsymbol{x} & 10.2 \mathrm{~cm} & 15.9 \mathrm{~cm} & 19.4 \mathrm{~cm} \\ \hline \text { Radius, } \boldsymbol{r} & 3.4 \mathrm{~cm} & 7.8 \mathrm{~cm} & 8.4 \mathrm{~cm} \\ \hline \end{array} $$ c. Rank the three cans in part (b) according to efficiency. Explain.

Describe a real-life situation that can be modeled by a rational equation. Justify your answer.

The total amount I (in millions of dollars) of healthcare expenditures and the residential population P (in millions) in the United States can be modeled by $$ \begin{aligned} &I=\frac{171,000 t+1,361,000}{1+0.018 t} \text { and } \\ &P=2.96 t+278.649 \end{aligned} $$ where \(t\) is the number of years since 2000. Find a model \(M\) for the annual healthcare expenditures per resident. Estimate the annual healthcare expenditures per resident in 2010.

In Exercises 39-44, simplify the complex fraction. \(\frac{\frac{x}{3}-6}{10+\frac{4}{x}}\)

Simplify the complex fraction. \(\frac{\frac{1}{2 x-5}-\frac{7}{8 x-20}}{\frac{x}{2 x-5}}\)

Your school purchases a math software program. The program has an initial cost of $$\$ 500$$ plus $$\$ 20$$ for each student that uses the program. (See Example 5.) a. Estimate how many students must use the program for the average cost per student to fall to \(\$ 30\). b. What happens to the average cost as more students use the program?

To join a rock climbing gym, you must pay an initial fee of $$\$ 100$$ and a monthly fee of $$\$ 59$$. a. Estimate how many months you must purchase a membership for the average cost per month to fall to $$\$ 69$$. b. What happens to the average cost as the number of months that you are a member increases? 43\. USING STRUCTURE