Functions and Their Graphs

Economics The participation rate is the number of people in the labor force divided by the civilian population (excludes military). Let $L\left(x\right)$ represent the size of the labor force in year x and $P\left(x\right)$represent the civilian population in year x. Determine a function that represents the participation rate $R$ as a function of x.

Crimes Suppose that $V\left(x\right)$ represents the number of violent crimes committed in year x and $P\left(x\right)$ represents the number of property crimes committed in year x. Determine a function $T$ that represents the combined total of violent crimes and property crimes in year x.

Health Care Suppose that $P\left(x\right)$ represents the percentage of income spent on health care in year $x$ and $I\left(x\right)$ represents income in year $x$. Determine a function $H$ that represents total health care expenditures in year $x$.

Income Tax Suppose that $I\left(x\right)$ represents the income of an individual in year $x$ before taxes and $T\left(x\right)$ represents the individual’s tax bill in year $x$. Determine a function $N$ that

represents the individual’s net income (income after taxes) in year $x$.

Profit Function Suppose that the revenue $R$, in dollars, from selling x cell phones, in hundreds, is $R\left(x\right)=-1.2x^2+220x$. The cost $C$, in dollars, of selling x cell phones is $C\left(x\right)=0.05x^3-2x^2+65x+500$.

(a) Find the profit function, $P\left(x\right)=R\left(x\right)-C\left(x\right)$.

(b) Find the profit if $x=15$ hundred cell phones are sold.

(c) Interpret $P\left(15\right)$.

Profit Function Suppose that the revenue $R$, in dollars, from selling x clocks is $R\left(x\right)=30x$. The cost $C$, in dollars, of selling x clocks is $C\left(x\right)=0.1x^2+7x+400$.

(a) Find the profit function, $P\left(x\right)=R\left(x\right)-C\left(x\right)$.

(b) Find the profit if $x=30$ clocks are sold.

(c) Interpret $P\left(30\right)$.

Some functions f have the property that $f(a+b)=f\left(a\right)+f\left(b\right)$ for all real numbers a and b. Which of the following functions have this property?

$$\begin{gathered} \left(a\right) h\left(x\right)=2x \left(b\right) g\left(x\right)=x^2 \\ \left(c\right) F\left(x\right)=5x-2 \left(d\right) G\left(x\right)=\frac{1}{x} \end{gathered}$$

Are the functions $f\left(x\right)=x-1$ and $g\left(x\right)=\frac{x^2-1}{x+1}$ the same? Explain.

Investigate when, historically, the use of the function notation $y=f\left(x\right)$ first appeared.

Find a function H that multiplies a number x by 3, then subtracts the cube of x and divides the result by your age.

The intercepts of the equation $x^2+4y^2=16$ are ______________________________. 

True or False: The point $\left(-2,-6\right)$ is on the graph of the equation $x=2y-2$. 

A set of points in the xy-plane is the graph of a function if and only if every __________ line intersects the graph in at most one point.

If the point $\left(5,-3\right)$ is a point on the graph of f, then$f\left(5\right)=$ ________.

Find a so that the point $\left(-1, 2\right)$ is on the graph of $f\left(x\right)=a{x}^2+4$. 

True or False: A function can have more than one y-intercept. 

True or False: The graph of a function $y=f\left(x\right)$ always crosses the y-axis. 

True or False: The y-intercept of the graph of the function $y=f\left(x\right)$, whose domain is all real numbers, is $f\left(0\right)$.

Use the given graph of the function $f$ to answer parts (a) – (n).

(a) Find $f\left(0\right)$ and $f\left(-6\right)$.

(b) Find $f\left(6\right)$ and $f\left(11\right)$.

(c) Is $f\left(3\right)$ positive or negative?

(d) Is $f\left(-4\right)$ positive or negative?

(e) For what values of $x$ is $f\left(x\right)=0$?

(f) For what values of $x$ is $f\left(x\right)>0$?

(g) What is the domain of $f$?

(h) What is the range of $f$?

(i) What are the x-intercepts?

(j) What is the y-intercept?

(k) How often does the line $y=\frac{1}{2}$ intersect the graph?

(l) How often does the line $x=5$ intersect the graph?

(m) For what values of $x$ does $f\left(x\right)=3$?

(n) For what values of $x$ does $f\left(x\right)=-2$? 

Use the given graph of the function f to answer parts (a) – (n).

(a) Find $f\left(0\right)$  and $f\left(6\right)$.

(b) Find $f\left(2\right)$ and $f\left(-2\right)$.

(c) Is $f\left(3\right)$ positive or negative?

(d) Is $f\left(-1\right)$ positive or negative?

(e) For what values of x is $f\left(x\right) = 0$?

(f) For what values of x is $f\left(x\right) <0$?

(g) What is the domain of f ?

(h) What is the range of f ?

(i) What are the x-intercepts?

(j) What is the y-intercept?

(k) How often does the line y = -1 intersect the graph?

(l) How often does the line x = 1 intersect the graph?

(m) For what value of x does $f\left(x\right) = 3$?

(n) For what value of x does $f\left(x\right) = -2$?

Determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis, the y-axis, or the origin.

Determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis, the y-axis, or the origin.

Determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis, the y-axis, or the origin.

Determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis, the y-axis, or the origin.

Determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis, the y-axis, or the origin.

Determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis, the y-axis, or the origin.

Determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis, the y-axis, or the origin.

Determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis, the y-axis, or the origin.

Determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis, the y-axis, or the origin.

Determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis, the y-axis, or the origin.

Determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis, the y-axis, or the origin.

Determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis, the y-axis, or the origin.

$f\left(x\right)=2x^2-x-1$

(a) Is the point $\left(-1, 2\right)$ on the graph of $f$?

(b) If $x=-2$, what is $f\left(x\right)$? What point is on the graph of $f$?

(c) If $f\left(x\right)=-1$, what is $x$? What point(s) are on the graph of $f$?

(d) What is the domain of $f$?

(e) List the x-intercepts, if any, of the graph of $f$.

(f) List the y-intercept, if there is one, of the graph of $f$. 

$f\left(x\right)=-3x^2+5x$

(a) Is the point $\left(-1, 2\right)$ on the graph of $f$?

(b) If $x=-2$, what is $f\left(x\right)$? What point is on the graph of $f$?

(c) If $f\left(x\right)=-2$, what is $x$? What point(s) are on the graph of $f$?

(d) What is the domain of $f$?

(e) List the x-intercepts, if any, of the graph of $f$.

(f) List the y-intercept, if there is one, of the graph of $f$.

$f\left(x\right)=\frac{x+2}{x-6}$

(a) Is the point $\left(3, 14\right)$ on the graph of $f$?

(b) If $x=4$, what is $f\left(x\right)$? What point is on the graph of $f$?

(c) If $f\left(x\right)=2$, what is $x$? What point(s) are on the graph of $f$?

(d) What is the domain of $f$?

(e) List the x-intercepts, if any, of the graph of $f$.

(f) List the y-intercept, if there is one, of the graph of $f$. 

$f\left(x\right)=\frac{x^2+2}{x+4}$

(a) Is the point $\left(1,\frac{3}{5}\right)$ on the graph of $f$?

(b) If $x=0$, what is $f\left(x\right)$? What point is on the graph of $f$?

(c) If $f\left(x\right)=\frac{1}{2}$, what is $x$? What point(s) are on the graph of $f$?

(d) What is the domain of $f$?

(e) List the x-intercepts, if any, of the graph of $f$.

(f) List the y-intercept, if there is one, of the graph of $f$. 

$f\left(x\right)=\frac{2x^2}{x^4+1}$

(a) Is the point $\left(-1, 1\right)$ on the graph of $f$?

(b) If $x=2$, what is $f\left(x\right)$? What point is on the graph of $f$?

(c) If $f\left(x\right)=1$, what is $x$? What point(s) are on the graph of $f$?

(d) What is the domain of $f$?

(e) List the x-intercepts, if any, of the graph of $f$.

(f) List the y-intercept, if there is one, of the graph of $f$. 

$f\left(x\right)=\frac{2x}{x-2}$

(a) Is the point $\left(\frac{1}{2},-\frac{2}{3}\right)$ on the graph of $f$?

(b) If $x=4$, what is $f\left(x\right)$? What point is on the graph of $f$?

(c) If $f\left(x\right)=1$, what is $x$? What point(s) are on the graph of $f$?

(d) What is the domain of $f$?

(e) List the x-intercepts, if any, of the graph of $f$.

(f) List the y-intercept, if there is one, of the graph of $f$. 

According to physicist Peter Brancazio, the key to a successful foul shot in basketball lies in the

arc of the shot. Brancazio determined the optimal angle of the arc from the free-throw line to be 45 degrees. The arc also depends on the velocity with which the ball is shot. If a player shoots a foul shot, releasing the ball at a 45-degree angle from a position 6 feet above the floor, then the path of

the ball can be modeled by the function

$h\left(x\right)=-\frac{44x^2}{v^2}+x+6$

where h is the height of the ball above the floor, x is the forward distance of the ball in front of the foul line, and v is the initial velocity with which the ball is shot in feet per second. Suppose a player shoots a ball with an initial velocity of 28 feet per second.

(a) Determine the height of the ball after it has traveled 8 feet in front of the foul line.

(b) Determine h(12). What does this value represent?

(c) Find additional points and graph the path of the basketball.

(d) The center of the hoop is 10 feet above the floor and 15 feet in front of the foul line. Will the ball go through the hoop? Why or why not? If not, with what initial velocity must the ball be shot in order for the ball to go through the hoop?

The last player in the NBA to use an underhand foul shot (a “granny” shot) was Hall of Fame

forward Rick Barry who retired in 1980. Barry believes that current NBA players could increase their free-throw percentage if they were to use an underhand shot. Since underhand shots are released from a lower position, the angle of the shot must be increased. If a player shoots an underhand foul shot, releasing the ball at a 70-degree angle from a position 3.5 feet above the floor, then the path of the ball can be modeled by the function h(x) = -

$h\left(x\right)=-\frac{136x^2}{v^2}+2.7x+3.5$

where h is the height of the ball above the floor, x is the forward distance of the ball in front of the foul line, and v is the initial velocity with which the ball is shot in feet per second.

(a) The center of the hoop is 10 feet above the floor and 15 feet in front of the foul line. Determine the initial velocity with which the ball must be shot in order for the ball to go through the hoop.

(b) Write the function for the path of the ball using the velocity found in part (a).

(c) Determine h(9). What does this value represent?

(d) Find additional points and graph the path of the basketball.

Motion of a Golf Ball A golf ball is hit with an initial velocity of 130 feet per second at an inclination of 45° to the horizontal. In physics, it is established that the height h of the golf ball is given by the function 

$h\left(x\right)=\frac{-32x^2}{{130}^2}+x$

where x is the horizontal distance that the golf ball has traveled. 

(a) Determine the height of the golf ball after it has traveled 100 feet. 

(b) What is the height after it has traveled 300 feet? 

(c) What is h(500)? Interpret this value. 

(d) How far was the golf ball hit? 

(e) Use a graphing utility to graph the function h = h(x). 

(f) Use a graphing utility to determine the distance that the ball has traveled when the height of the ball is 90 feet. 

(g) Create a TABLE with TblStart = 0 and $∆Tbl$ = 25. To the nearest 25 feet, how far does the ball travel before it reaches a maximum height? What is the maximum height?

(h) Adjust the value of $∆Tbl$ until you determine the distance, to within 1 foot, that the ball travels before it reaches a maximum height.

Cross-sectional Area The cross-sectional area of a beam

cut from a log with radius 1 foot is given by the function

$A\left(x\right)=4x\sqrt{1-x^2}$, where x represents the length, in feet,

of half the base of the beam. See the figure.

(a) Find the domain of A.

(b) Use a graphing utility to graph the function A = A(x).

(c) Create a TABLE with TblStart = 0 and Tbl = 0.1. 

Which value of x in the domain found in part (a) maximizes the cross sectional area? What should be the length of the base of the beam to maximize the cross-sectional area?

Cost of Trans-Atlantic Travel A Boeing 747 crosses the Atlantic Ocean (3000 miles) with an airspeed of 500 miles per hour. The cost C (in dollars) per passenger is given by -

$C\left(x\right)=100+\frac{x}{10}+\frac{36000}{x}$

Where x is the ground speed $(airspeed + wind).$

(a). Use a graphing utility to graph the function C = C(x).

(b). Create a TABLE with TblStart = 0 and $\Delta Tbl=50.$

(c). To the nearest 50 miles per hour, what ground speed minimizes the cost per passenger?

Effect of Elevation on Weight If an object weighs m pounds at sea level, then its weight W (in pounds) at a height of h miles above sea level is given approximately by 

$W\left(h\right)=m{\left(\frac{4000}{4000+h}\right)}^2$

(a) If Amy weighs 120 pounds at sea level, how much will she weigh on Pike’s Peak, which is 14,110 feet above sea level? 

(b) Use a graphing utility to graph the function W = W(h). Use m = 120 pounds. 

(c) Create a TABLE with TblStart = 0 and $∆Tbl$ = 0.5 to see how the weight W varies as h changes from 0 to 5 miles. 

(d) At what height will Amy weigh 119.95 pounds? 

(e) Does your answer to part (d) seem reasonable? Explain. 

The graph of two functions, f and g, is illustrated. 

Use the graph to answer parts (a) – (f).

$$\begin{gathered} \left(a\right) (f + g)\left(2\right) \left(b\right) (f + g)\left(4\right) \\ \left(c\right) (f - g)\left(6\right) \left(d\right) (g-f)\left(6\right) \\ \left(e\right) (f · g)\left(2\right) \left(f\right) \left(\frac{f}{g}\right)\left(4\right) \end{gathered}$$

Reading and Interpreting Graphs Let C be the function whose graph is given in the next column. This graph represents the cost C of manufacturing q computers in a day.

(a) Determine C(0). Interpret this value.

(b) Determine C(10). Interpret this value.

(c) Determine C(50). Interpret this value.

(d) What is the domain of C? What does this domain imply in terms of daily production?

(e) Describe the shape of the graph.

(f) The point (30, 32000) is called an inflection point. Describe the behavior of the graph around the inflection point.

Reading and Interpreting Graphs Let C be the function whose graph is given below. This graph represents the cost C of using m anytime cell phone minutes in a month for a five-person family plan.

(a) Determine C(0). Interpret this value.

(b) Determine C(1000). Interpret this value.

(c) Determine C(2000). Interpret this value.

(d) What is the domain of C? What does this domain imply in terms of the number of anytime minutes?

(e) Describe the shape of the graph.

Describe how you would proceed to find the domain and range of a function if you were given its graph. How would your strategy change if you were given the equation defining the function instead of its graph?

How many x-intercepts can the graph of a function have? How many y-intercepts can the graph of a function have?

Is a graph that consists of a single point the graph of a function? Can you write the equation of such a function?