Chapter 2

Show that $$ f(x)=\frac{3 x-2}{5 x-3} $$ is its own inverse.

Begin by graphing the standard cubic function, \(f(x)-x^{3} .\) Then use transformations of this graph to graph the given function. $$ g(x)-x^{3}-2 $$

Find the area of the donut-shaped region bounded by the graphs of \((x-2)^{2}+(y+3)^{2}=25\) and \((x-2)^{2}+(y+3)^{2}=36\).

Begin by graphing the standard cubic function, \(f(x)-x^{3} .\) Then use transformations of this graph to graph the given function. $$ g(x)-(x-3)^{3} $$

Explain how to use the general form of a line's equation to find the line's slope and \(y\) -intercept.

A tangent line to a circle is a line that intersects the circle at exactly one point. The tangent line is perpendicular to the radius of the circle at this point of contact. Write an equation in point-slope form for the line tangent to the circle whose equation is \(x^{2}+y^{2}=25\) at the point \((3,-4)\)

If \(f(2)=6,\) and \(f\) is one-to-one, find \(x\) satisfying \(8+f^{-1}(x-1)=10\)

Begin by graphing the standard cubic function, \(f(x)-x^{3} .\) Then use transformations of this graph to graph the given function. $$ g(x)-(x-2)^{3} $$

Explain how to use intercepts to graph the general form oE a line's equation.

Solve each quadratic equation by the method of your choice. $$ 0=-2(x-3)^{2}+8 $$

Furry Finances A pet insurance policy has a monthly rate that is a function of the age of the insured dog or cat. For pets whose age does not exceed \(4,\) the monthly cost is \(\$ 20\). The cost then increases by \(\$ 2\) for each successive year of the pet's age. $$\begin{array}{|c|c|}\hline\hline\text { Age Not Exceeding } & \text { Monthly Cost } \\\\\hline 4 & \$ 20 \\\5 & \$ 22 \\\6 & \$ 24\end{array}$$ The cost schedule continues in this manner for ages not exceeding \(10 .\) The cost for pets whose ages exceed 10 is S40. Use this information to create a graph that shows the monthly cost of the insurance, \(f(x)\), for a pet of age \(x,\) where the function's domain is \([0,14]\).

In Tom Stoppard's play Arcadia , the characters dream and talk about mathematics, including ideas involving graphing. composite functions, symmetry, and lack of symmetry in things that are tangled, mysterious, and unpredictable. Group members should read the play. Present a report on the ideas discussed by the characters that are related to concepts that we studied in this chapter. Bring in a copy of the play and read appropriate excerpts.

Begin by graphing the standard cubic function, \(f(x)-x^{3} .\) Then use transformations of this graph to graph the given function. $$ h(x)=-x^{3} $$

Solve each quadratic equation by the method of your choice. $$ -x^{2}-2 x+1=0 $$

What does it mean if a function \(f\) is increasing on an interval?

Will help you prepare for the material covered in the next section. Let \(\left(x_{1}, y_{1}\right)=(7,2) \quad\) and \(\quad\left(x_{2}, y_{2}\right)=(1,-1) . \quad\) Find \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}} .\) Express the answer in simplified radical form.

Use the graph of \(f(x)=x^{2}\) to graph \(g(x)=(x+3)^{2}+1\).

The bar graph shows the population of the United States, in millions, for six selected years. (GRAPH CANNOT COPY Here are two functions that model the data: (IMAGE CANNOT COPY) Use the functions to solve A department store has two locations in a city. From 2008 through \(2012,\) the profits for each of the store's two branches are modeled by the functions \(f(x)=-0.44 x+13.62\) and \(g(x)-0.51 x+11.14 .\) In each model, \(x\) represents the number of years after 2008 , and \(f\) and \(g\) represent the profit, in millions of dollars. a. What is the slope of \(f ?\) Describe what this means. b. What is the slope of \(g\) ? Describe what this means. c. Find \(f+g .\) What is the slope of this function? What does this mean?

Suppose that a function \(f\) whose graph contains no breaks or gaps on \((a, c)\) is increasing on \((a, b),\) decreasing on \((b, c)\) and defined at \(b\). Describe what occurs at \(x-b\). What does the function value \(f(b)\) represent?

Will help you prepare for the material covered in the next section. Use a rectangular coordinate system to graph the circle with center \((1,-1)\) and radius 1.

Begin by graphing the standard cubic function, \(f(x)-x^{3} .\) Then use transformations of this graph to graph the given function. $$ h(x)-\frac{1}{2} x^{3} $$

If you are given a function's equation, how do you determine if the function is even, odd, or neither?

Will help you prepare for the material covered in the next section. Solve by completing the square: \(y^{2}-6 y-4=0\)

Begin by graphing the standard cubic function, \(f(x)-x^{3} .\) Then use transformations of this graph to graph the given function. $$ h(x)-\frac 12 x^{3} $$

If you are given a function's graph, how do you determine if the function is even, odd, or neither?

Begin by graphing the standard cubic function, \(f(x)-x^{3} .\) Then use transformations of this graph to graph the given function. $$ r(x)-(x-3)^{3}+2 $$

If a function is defined by an equation, explain how to find its domain.

What is a piecewise function?

Begin by graphing the standard cubic function, \(f(x)-x^{3} .\) Then use transformations of this graph to graph the given function. $$ r(x)-(x-2)^{3}+1 $$

If equations for \(f\) and \(g\) are given, explain how to find \(f-g .\)

Explain how to find the difference quotient of a function \(f\) \(\frac{f(x+h)-f(x)}{h},\) if an equation for \(f\) is given.

Begin by graphing the standard cubic function, \(f(x)-x^{3} .\) Then use transformations of this graph to graph the given function. $$ h(x)-\frac 12(x-3)^{3}-2 $$

If equations for two functions are given, explain how to obtain the quotient function and its domain.

The function $$ f(x)=-0.00002 x^{3}+0.008 x^{2}-0.3 x+6.95$$ models the number of annual physician visits, \(f(x),\) by a person of age \(x .\) Graph the function in a \([0,100,5]\) by \([0,40,2]\) viewing rectangle. What does the shape of the graph indicate about the relationship between one's age and the number of annual physician visits? Use the TABLE or minimum function capability to find the coordinates of the minimum point on the graph of the function. What does this mean?

Begin by graphing the standard cubic function, \(f(x)-x^{3} .\) Then use transformations of this graph to graph the given function. $$ h(x)-\frac 1 2(x-2)^{3}-1 $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. A linear function that models tuition and fees at public four-year colleges from 2000 through 2012 has negative slope.

Describe a procedure for finding \((f \circ g)(x) .\) What is the name of this function?

Use a graphing utility to graph each function. Use a \([-5,5,1]\) by \([-5,5,1]\) viewing rectangle. Then find the intervals on which the function is increasing, decreasing, or constant. $$f(x)=x^{3}-6 x^{2}+9 x+1$$

Begin by graphing the cube root function, \(f(x)-\sqrt[3]{x} .\) Then use transformations of this graph to graph the given function. $$ g(x)=\sqrt[3]{x}+2 $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because the variable \(m\) does not appear in \(A x+B y+C-0,\) equations in this form make it impossible to determine the line's slope.

Use a graphing utility to graph each function. Use a \([-5,5,1]\) by \([-5,5,1]\) viewing rectangle. Then find the intervals on which the function is increasing, decreasing, or constant. $$g(x)=\left|4-x^{2}\right|$$

Begin by graphing the cube root function, \(f(x)-\sqrt[3]{x} .\) Then use transformations of this graph to graph the given function. $$ g(x)-\sqrt[3]{x}-2 $$

Use a graphing utility to graph each function. Use a \([-5,5,1]\) by \([-5,5,1]\) viewing rectangle. Then find the intervals on which the function is increasing, decreasing, or constant. $$h(x)=|x-2|+|x+2|$$

Begin by graphing the cube root function, \(f(x)-\sqrt[3]{x} .\) Then use transformations of this graph to graph the given function. $$ g(x)-\sqrt[3]{x+2} $$

Use a graphing utility to graph each function. Use a \([-5,5,1]\) by \([-5,5,1]\) viewing rectangle. Then find the intervals on which the function is increasing, decreasing, or constant. $$f(x)=x^{3}(x-4)$$

Begin by graphing the cube root function, \(f(x)-\sqrt[3]{x} .\) Then use transformations of this graph to graph the given function. $$ g(x)-\sqrt[3]{x-2} $$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Every line in the rectangular coordinate system has an equation that can be expressed in slope-intercept form.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I used a function to model data from 1980 through 2005 . The independent variable in my model represented the number of years after \(1980,\) so the function's domain was \(|x| x-0,1,2,3, \ldots, 25\\}\)

Use a graphing utility to graph each function. Use a \([-5,5,1]\) by \([-5,5,1]\) viewing rectangle. Then find the intervals on which the function is increasing, decreasing, or constant. $$g(x)=x^{\frac{2}{3}}$$

Begin by graphing the cube root function, \(f(x)-\sqrt[3]{x} .\) Then use transformations of this graph to graph the given function. $$ h(x)-\frac{1}{2} \sqrt[3]{x+2} $$