Chapter 7

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

\(h(x)=2 \ln (x+9)\)

Determine whether the inverse of \(f\) is a function. Then find the inverse. $$f(x)=\frac{3}{x}-2$$

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 \(f(x)=\frac{a}{x}\). $$ g(x)=\frac{9 x-3}{x+7} $$

Simplify the complex fraction. \(\frac{15-\frac{2}{x}}{\frac{x}{5}+4}\)

Determine whether the inverse of \(f\) is a function. Then find the inverse. $$f(x)=\frac{5}{x}-6$$

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?

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

Complete the table for the function \(y=\frac{x+4}{x^2-16}\) Then use the trace feature of a graphing calculator to explain the behavior of the function at x = ?4. $$ \begin{array}{|c|c|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline-3.5 & \\ \hline-3.8 & \\ \hline-3.9 & \\ \hline-4.1 & \\ \hline-4.2 & \\ \hline \end{array} $$

Determine whether the inverse of \(f\) is a function. Then find the inverse. $$f(x)=\frac{4}{11-2 x}$$

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

Simplify the complex fraction. \(\frac{\frac{16}{x-2}}{\frac{4}{x+1}+\frac{6}{x}}\)

You and your friend are asked to state the domain of the expression below. $$ \frac{x^2+6 x-27}{x^2+4 x-45} $$ Your friend claims the domain is all real numbers except 5. You claim the domain is all real numbers except ?9 and 5. Who is correct? Explain.

Determine whether the inverse of \(f\) is a function. Then find the inverse. $$f(x)=\frac{8}{9+5 x}$$

What is the vertical asymptote of the graph of the function \(y=\frac{2}{x+4}+7\) ? (A) \(x=-7\) (B) \(x=-4\) (C) \(x=4\) (D) \(x=7\)

Simplify the complex fraction. \(\frac{\frac{1}{3 x^2-3}}{\frac{5}{x+1}-\frac{x+4}{x^2-3 x-4}}\)

Determine whether the inverse of \(f\) is a function. Then find the inverse. $$f(x)=\frac{1}{x^2}+4$$

What are the \(x\)-intercept(s) of the graph of the function \(y=\frac{x-5}{x^2-1}\) ? (A) \(1,-1\) (B) 5 (C) 1 (D) \(-5\)

Simplify the complex fraction. \(\frac{\frac{3}{x-2}-\frac{6}{x^2-4}}{\frac{3}{x+2}+\frac{1}{x-2}}\)

Determine whether the inverse of \(f\) is a function. Then find the inverse. $$f(x)=\frac{1}{x^4}-7$$

The time \(t\) (in seconds) it takes for sound to travel 1 kilometer can be modeled by $$ t=\frac{1000}{0.6 T+331} $$ where \(T\) is the air temperature (in degrees Celsius). a. You are 1 kilometer from a lightning strike. You hear the thunder \(2.9\) seconds later. Use a graph to find the approximate air temperature. b. Find the average rate of change in the time it takes sound to travel 1 kilometer as the air temperature increases from \(0^{\circ} \mathrm{C}\) to \(10^{\circ} \mathrm{C}\).

Perform the indicated operations. $$ \frac{2 x^2+x-15}{2 x^2-11 x-21} \cdot(6 x+9) \div \frac{2 x-5}{3 x-21} $$

A business is studying the cost to remove a pollutant from the ground at its site. The function \(y=\frac{15 x}{1.1-x}\) models the estimated cost \(y\) (in thousands of dollars) to remove \(x\) percent (expressed as a decimal) of the pollutant. a. Graph the function. Describe a reasonable domain and range. b. How much does it cost to remove \(20 \%\) of the pollutant? \(40 \%\) of the pollutant? \(80 \%\) of the pollutant? Does doubling the percentage of the pollutant removed double the cost? Explain.

Perform the indicated operations. $$ \left(x^3+8\right) \cdot \frac{x-2}{x^2-2 x+4} \div \frac{x^2-4}{x-6} $$

The recommended percent \(p\) of nitrogen (by volume) in the air that a diver breathes is given by \(p=\frac{105.07}{d+33}\), where \(d\) is the depth (in feet) of the diver. Find the depth when the air contains \(47 \%\) recommended nitrogen by (a) solving an equation, and (b) using the inverse of the function.

In Exercises 47-50, use a graphing calculator to graph the function. Then determine whether the function is even, odd, or neither. $$ h(x)=\frac{6}{x^2+1} $$

You plan a trip that involves a 40 -mile bus ride and a train ride. The entire trip is 140 miles. The time (in hours) the bus travels is \(y_1=\frac{40}{x}\), where \(x\) is the average speed (in miles per hour) of the bus. The time (in hours) the train travels is \(y_2=\frac{100}{x+30}\). Write and simplify a model that shows the total time \(y\) of the trip.

Use a graphing calculator to determine where \(f(x)=g(x)\). $$f(x)=\frac{2}{3 x}, g(x)=x$$

In Exercises 47-50, use a graphing calculator to graph the function. Then determine whether the function is even, odd, or neither. $$ f(x)=\frac{2 x^2}{x^2-9} $$

Is it possible to write two rational functions whose product when graphed is a parabola and whose quotient when graphed is a hyperbola? Justify your answer.

Use a graphing calculator to determine where \(f(x)=g(x)\). $$f(x)=-\frac{3}{5 x}, g(x)=-x$$

In Exercises 47-50, use a graphing calculator to graph the function. Then determine whether the function is even, odd, or neither. $$ y=\frac{x^3}{3 x^2+x^4} $$

Your friend claims that the least common multiple of two numbers is always greater than each of the numbers. Is your friend correct? Justify your answer.

Find two rational functions f and g that have the stated product and quotient. $$ (f g)(x)=x^2,\left(\frac{f}{g}\right)(x)=\frac{(x-1)^2}{(x+2)^2} $$

Use a graphing calculator to determine where \(f(x)=g(x)\). $$f(x)=\frac{1}{x}+1, g(x)=x^2$$

In Exercises 47-50, use a graphing calculator to graph the function. Then determine whether the function is even, odd, or neither. $$ f(x)=\frac{4 x^2}{2 x^3-x} $$

Solve the equation. Check your solution. $$ \frac{1}{2} x+4=\frac{3}{2} x+5 $$

Use a graphing calculator to determine where \(f(x)=g(x)\). $$f(x)=\frac{2}{x}+1, g(x)=x^2+1$$

Your friend claims it is possible for a rational function to have two vertical asymptotes. Is your friend correct? Justify your answer.

You borrow \(P\) dollars to buy a car and agree to repay the loan over \(t\) years at a monthly interest rate of \(i\) (expressed as a decimal). Your monthly payment \(M\) is given by either formula below. $$ M=\frac{P i}{1-\left(\frac{1}{1+i}\right)^{12 t}} \quad \text { or } \quad M=\frac{P i(1+i)^{12 t}}{(1+i)^{12 t}-1} $$ a. Show that the formulas are equivalent by simplifying the first formula. b. Find your monthly payment when you borrow \(\$ 15,500\) at a monthly interest rate of \(0.5 \%\) and repay the loan over 4 years.

Solve the equation. Check your solution. $$ \frac{1}{3} x-2=\frac{3}{4} x $$

Golden rectangles are rectangles for which the ratio of the width \(w\) to th length \(\ell\) is equal to the ratio of \(\ell\) to \(\ell+w\). The ratio of the length to the width for these rectangles is called the golden ratio. Find the value of the golden ratio using a rectangle with a width of 1 unit.

Is it possible to write two rational functions whose sum is a quadratic function? Justify your answer.

Solve the equation. Check your solution. $$ \frac{1}{4} x-\frac{3}{5}=\frac{9}{2} x-\frac{4}{5} $$

In what line(s) is the graph of \(y=\frac{1}{x}\) symmetric? What does this symmetry tell you about the inverse of the function \(f(x)=\frac{1}{x}\) ?

Use technology to rewrite the function \(g(x)=\frac{(97.6)(0.024)+x(0.003)}{12.2+x}\) in the form \(f(x)=\frac{a}{x-h}+k\). Describe the graph of \(g\) as a transformation of the graph of \(f(x)=\frac{a}{x}\).

Solve the equation. Check your solution. $$ \frac{1}{2} x+\frac{1}{3}=\frac{3}{4} x-\frac{1}{5} $$

Find the inverse of the function. (Hint: Try rewriting the function by using either inspection or long division.) $$f(x)=\frac{3 x+1}{x-4}$$

Write the prime factorization of the number. If the number is prime, then write prime. 42

Find the inverse of the function. (Hint: Try rewriting the function by using either inspection or long division.) $$f(x)=\frac{4 x-7}{2 x+3}$$