Section 5.7 Chapter 5 Summary and Review
Subsection 5.7.1 Glossary
function
input variable
output variable
cube root
absolute value
proportional
direct variation
inverse variation
constant of variation
concavity
scaling
horizontal asymptote
vertical asymptote
Subsection 5.7.2 Key Concepts
A function can be described in words, by a table, by a graph, or by an equation.
Function Notation.
Finding the value of the output variable that corresponds to a particular value of the input variable is called evaluating the function.
The point \((a, b)\) lies on the graph of the function \(f\) if and only if \(f(a)=b\text{.}\)
Each point on the graph of the function \(f\) has coordinates \((x, f(x))\) for some value of \(x\text{.}\)
The Vertical Line Test.
A graph represents a function if and only if every vertical line intersects the graph in at most one point.
We can use a graphical technique to solve equations and inequalities.
\(b\) is the cube root of \(a\) if \(b\) cubed equals \(a\text{.}\) In symbols, we write
\begin{equation*} \blert{b=\sqrt[3]{a}~~~~\text{if}~~~~b^3=a} \end{equation*}Absolute Value.
The absolute value of \(x\) is defined by
\begin{equation*} \abs{x} = \begin{cases} x \amp \text{if } x\ge 0\\ x \amp \text{if } x\lt 0 \end{cases} \end{equation*}Absolute value bars act like grouping devices in the order of operations: you should complete any operations that appear inside absolute value bars before you compute the absolute value.
The maximum or minimum of a quadratic function occurs at the vertex.

Eight basic functions and their graphs are important in applications:
\begin{gather*} f(x)=x~~~~~~f(x)=\abs{x}~~~~~~f(x)=x^2~~~~~~f(x)=x^3\\ f(x)=\sqrt{x}~~~~~f(x)=\sqrt[3]{x}~~~~~~f(x)=\dfrac{1}{x}~~~~~~f(x)=\dfrac{1}{x^2} \end{gather*} Two variables are directly proportional if the ratios of their corresponding values are always equal.
Direct Variation.
\(y\) varies directly with \(x\) if
\begin{equation*} y = kx \end{equation*}where \(k\) is a positive constant called the constant of variation.
Direct variation defines a linear function of the form
\begin{equation*} y = f (x) = kx \end{equation*}The positive constant \(k\) in the equation \(y = kx\) is just the slope of the graph.Direct variation has the following scaling property: increasing \(x\) by any factor causes \(y\) to increase by the same factor.
Direct Variation with a Power.
\(y\) varies directly with a power of \(x\) if
\begin{equation*} y = kx^n \end{equation*}where \(k\) and \(n\) are positive constants.
If the ratio \(\dfrac{y}{x^n}\) is constant, then \(y\) varies directly with \(x^n\text{.}\)
Inverse Variation.
\(y\) varies inversely with \(x\) if
\begin{equation*} y = \dfrac{k}{x}\text{, }x \ne 0 \end{equation*}where \(k\) is a positive constant.
Inverse Variation with a Power.
\(y\) varies inversely with \(x^n\) if
\begin{equation*} y = \frac{k}{x^n}\text{, }x \ne 0 \end{equation*}where \(k\) and \(n\) are positive constants.
If the product \(~yx^n~\) is constant and \(n\) is positive, then \(y\) varies inversely with \(x^n\text{.}\)
A graph that bends upward is called concave up, and one that bends downward is concave down.
Exercises 5.7.3 Chapter 5 Review Problems
Exercise Group.
Which of the tables in Problems 1–4 describe functions? Why or why not?
1.
\(x\)  \(2\)  \(1\)  \(0\)  \(1\)  \(2\)  \(3\) 
\(y\)  \(6\)  \(0\)  \(1\)  \(2\)  \(6\)  \(8\) 
A function: Each \(x\) has exactly one associated \(y\)value.
2.
\(p\)  \(3\)  \(3\)  \(2\)  \(2\)  \(2\)  \(0\) 
\(q\)  \(2\)  \(1\)  \(4\)  \(4\)  \(3\)  \(0\) 
Not a function
3.
Student  Score on IQ test 
Score on SAT test 
(A)  \(118\)  \(649\) 
(B)  \(98\)  \(450\) 
(C)  \(110\)  \(590\) 
(D)  \(105\)  \(520\) 
(E)  \(98\)  \(490\) 
(F)  \(122\)  \(680\) 
Not a function: The IQ of \(98\) has two possible SAT scores.
4.
Student  Correct answers on math quiz 
Quiz grade 
(A)  \(13\)  \(85\) 
(B)  \(15\)  \(89\) 
(C)  \(10\)  \(79\) 
(D)  \(12\)  \(82\) 
(E)  \(16\)  \(91\) 
(F)  \(18\)  \(95\) 
A function
5.
The total number of barrels of oil pumped by the AQ oil company is given by the formula
where \(N\) is the number of barrels of oil \(t\) days after a new well is opened. Evaluate \(N(10)\) and explain what it means.
\(N(10) = 7000\text{:}\) Ten days after the new well is opened, the company has pumped a total of \(7000\) barrels of oil.
6.
The number of hours required for a boat to travel upstream between two cities is given by the formula
where \(v\) represents the boat's top speed in miles per hour. Evaluate \(H(16)\) and explain what it means.
\(H(16) = 3\text{:}\) At 16 mph, the trip takes 3 hours.
Exercise Group.
For Problems 710, evaluate the function for the given values.
7.
\(F(t)=\sqrt{1+4t^2}\text{,}\) \(~~F(0)~~\) and \(~~F(3)\)
\(F(0) = 1, ~~F(3) =\sqrt{37}\)
8.
\(G(x)=\sqrt[3]{x8}\text{,}\) \(~~G(0)~~\) and \(~~G(20)\)
\(G(0) = 2, ~~G(20) =\sqrt[3]{12}\)
9.
\(h(v)=6\abs{42v} \text{,}\) \(~~h(8)~~\) and \(~~h(8)\)
\(h(8) = 6, ~~h(8) = 14\)
10.
\(m(p)=\dfrac{120}{p+15} \text{,}\) \(~~m(5)~~\) and \(~~m(40)\)
\(m(5) = 6, ~~m(40) =4.8\)
11.
\(P(x)=x^26x+5\)
Compute \(P(0)\text{.}\)
Find all values of \(x\) for which \(P(x)=0\text{.}\)
\(\displaystyle P(0)=5\)
x=5,~x=1
12.
\(R(x)=\sqrt{4x^2}\)
Compute \(R(0)\text{.}\)
Find all values of \(x\) for which \(R(x)=0\text{.}\)
R(0)=2
\(\displaystyle x=2,~x=2\)
Exercise Group.
For Problems 13 and 14, refer to the graphs to answer the questions.
13.
Find \(f (2)\) and \(f (2)\text{.}\)
For what value(s) of \(t\) is \(f (t) = 4\text{?}\)
Find the \(t\) and \(f(t)\)intercepts of the graph.
What is the maximum value of \(f\text{?}\) For what value(s) of \(t\) does \(f\) take on its maximum value?
\(\displaystyle f (2) = 3, ~~f (2) = 5\)
\(\displaystyle t = 1, ~~t = 3\)
\(t\)intercepts \((3, 0), (4, 0)\text{;}\) \(f (t)\)intercept: \((0, 2)\)
Maximum value of \(5\) occurs at \(t = 2\)
14.
Find \(P(3)\) and \(P(3)\text{.}\)
For what value(s) of \(z\) is \(P(z) = 2\text{?}\)
Find the \(z\) and \(P(z)\)intercepts of the graph.
What is the minimum value of \(P\text{?}\) For what value(s) of \(z\) does \(P\) take on its minimum value?
\(\displaystyle P(3)=2, ~~P(3)=3\)
\(\displaystyle z = 5,~ \dfrac{1}{2},~4\)
\((4, 0), (1, 0), (5,0)\text{;}\) \((0, 3)\)
Maximum value of \(3\) occurs at \(z = 2\)
Exercise Group.
Which of the graphs in Problems 15–18 represent functions?
Exercise Group.
For Problems 19–22, graph the function by hand.
Exercise Group.
For Problems 23–26, graph the given function on a graphing calculator. Then use the graph to solve the equations and inequalities. Round your answers to one decimal place if necessary.
23.
\(y=\sqrt[3]{x} \)
Solve \(\sqrt[3]{x} = 0.8\)
Solve \(\sqrt[3]{x} = 1.5\)
Solve \(\sqrt[3]{x}\gt 1.7 \)
Solve \(\sqrt[3]{x}\le 1.26 \)
\(\displaystyle x = \dfrac{1}{2}= 0.5\)
\(\displaystyle x = \dfrac{27}{8}\approx 3.4\)
\(\displaystyle x \gt 4.9\)
\(\displaystyle x \le 2.0\)
24.
\(y=\dfrac{1}{x} \)
Solve \(\dfrac{1}{x} = 2.5\)
Solve \(\dfrac{1}{x} = 0.3125\)
Solve \(\dfrac{1}{x}\ge 0.\overline{2} \)
Solve \(\dfrac{1}{x}\lt 5\)
\(\displaystyle x = 0.4\)
\(\displaystyle x = 3.2\)
\(\displaystyle 0 \lt x \le 4.5\)
\(x \lt 0\) or \(x \gt 0.2\)
25.
\(y=\dfrac{1}{x^2} \)
Solve \(\dfrac{1}{x^2} = 0.03\)
Solve \(\dfrac{1}{x^2} = 6.25\)
Solve \(\dfrac{1}{x^2}\gt 0.16 \)
Solve \(\dfrac{1}{x^2}\le 4\)
\(\displaystyle x\approx\pm 5.8 \)
\(\displaystyle x = \pm 0.4\)
\(2.5\lt x \lt 0\) or \(0\lt x\lt 2.5\)
\(x \le 0.5\) or \(x\ge 0.5\)
26.
\(y=\sqrt{x} \)
Solve \(\sqrt{x} = 0.707\)
Solve \(\sqrt{x} = 1.7\)
Solve \(\sqrt{x}\lt 1.5 \)
Solve \(\sqrt{x}\ge 1.3 \)
\(\displaystyle x = 0.5\)
\(\displaystyle x = 2.9\)
\(\displaystyle 0 \le x \lt 2.3\)
\(\displaystyle x \ge 1.7\)
Exercise Group.
In Problems 27–30, \(y\) varies directly or inversely with a power of \(x\text{.}\) Find the power of \(x\) and the constant of variation, \(k\text{.}\) Write a formula for each function of the form \(y = kx^n\) or \(y = \dfrac{k}{x^n}\text{.}\)
27.
\(x\)  \(y\) 
\(2\)  \(4.8\) 
\(5\)  \(30.0\) 
\(8\)  \(76.8\) 
\(11\)  \(145.2\) 
\(y = 1.2x^2\)
28.
\(x\)  \(y\) 
\(1.4\)  \(75.6\) 
\(2.3\)  \(124.2\) 
\(5.9\)  \(318.6\) 
\(8.3\)  \(448.2\) 
\(y = 54x\)
29.
\(x\)  \(y\) 
\(0.5\)  \(40.0\) 
\(2.0\)  \(10.0\) 
\(4.0\)  \(5.0\) 
\(8.0\)  \(2.5\) 
\(y =\dfrac{20}{x} \)
30.
\(x\)  \(y\) 
\(1.5\)  \(320.0\) 
\(2.5\)  \(115.2\) 
\(4.0\)  \(45.0\) 
\(6.0\)  \(20.0\) 
\(y =\dfrac{720}{x^2} \)
31.
The distance s a pebble falls through a thick liquid varies directly with the square of the length of time \(t\) it falls.
If the pebble falls 28 centimeters in 4 seconds, express the distance it will fall as a function of time.
Find the distance the pebble will fall in \(6\) seconds.
\(\displaystyle d = 1.75t^2\)
63 cm
32.
The volume, \(V\text{,}\) of a gas varies directly with the temperature, \(T\text{,}\) and inversely with the pressure, \(P\text{,}\) of the gas.
If \(V = 40\) when \(T = 300\) and \(P = 30\text{,}\) express the volume of the gas as a function of the temperature and pressure of the gas.
Find the volume when \(T = 320\) and \(P = 40\text{.}\)
\(\displaystyle V=\dfrac{4T}{P}\)
\(\displaystyle 32\)
33.
The demand for bottled water is inversely proportional to the price per bottle. If Droplets can sell 600 bottles at $8 each, how many bottles can the company sell at $10 each?
\(480\) bottles
34.
The intensity of illumination from a light source varies inversely with the square of the distance from the source. If a reading lamp has an intensity of 100 lumens at a distance of 3 feet, what is its intensity 8 feet away?
14.0625 lumens
35.
A person's weight, \(w\text{,}\) varies inversely with the square of his or her distance, \(r\text{,}\) from the center of the Earth.
Express \(w\) as a function of \(r\text{.}\) Let \(k\) stand for the constant of variation.
Make a rough graph of your function.
How far from the center of the Earth must Neil be in order to weigh onethird of his weight on the surface? The radius of the Earth is about 3960 miles.
\(\displaystyle w = \dfrac{k}{r^2}\)
\(3960\sqrt{3}\approx 6860\) miles
36.
The period, \(T\text{,}\) of a pendulum varies directly with the square root of its length, \(L\text{.}\)
Express \(T\) as a function of \(L\text{.}\) Let \(k\) stand for the constant of variation.
Make a rough graph of your function.
If a certain pendulum is replaced by a new one fourfifths as long as the old one, what happens to the period?
Exercise Group.
For Problems 37 and 38, sketch a graph to illustrate the situations.
37.
Inga runs hot water into the bathtub until it is about half full. Because the water is too hot, she lets it sit for a while before getting into the tub. After several minutes of bathing, she gets out and drains the tub. Graph the water level in the bathtub as a function of time, from the moment Inga starts filling the tub until it is drained.
38.
David turns on the oven and it heats up steadily until the proper baking temperature is reached. The oven maintains that temperature during the time David bakes a pot roast. When he turns the oven off, David leaves the oven door open for a few minutes, and the temperature drops fairly rapidly during that time. After David closes the door, the temperature continues to drop, but at a much slower rate. Graph the temperature of the oven as a function of time, from the moment David first turns on the oven until shortly after David closes the door when the oven is cooling.
Exercise Group.
For Problems 39–42, sketch a graph by hand for the function.
39.
\(y\) varies directly with \(x^2\text{.}\) The constant of variation is \(k=0.25\text{.}\)
40.
\(y\) varies directly with \(x\text{.}\) The constant of variation is \(k = 1.5\text{.}\)
41.
\(y\) varies inversely with \(x\text{.}\) The constant of variation is \(k = 2\text{.}\)
42.
\(y\) varies inversely with \(x^2\text{.}\) The constant of variation is \(k = 4\text{.}\)
Exercise Group.
In Problems 43 and 44,
Plot the points and sketch a smooth curve through them.
Use your graph to discover the equation that describes the function.
Exercise Group.
In Problems 45–50,
Use the graph to complete the table of values.
By finding a pattern in the table of values, write an equation for the graph.
45.
\(x\)  \(0\)  \(4\)  \(8\)  \(\hphantom{000}\)  \(16\)  \(\hphantom{000}\) 
\(y\)  \(\hphantom{000}\)  \(\hphantom{000}\)  \(\hphantom{000}\)  \(10\)  \(\hphantom{000}\)  \(2\) 
\(x\) \(0\) \(4\) \(8\) \(14\) \(16\) \(22\) \(y\) \(24\) \(20\) \(16\) \(10\) \(8\) \(2\) \(\displaystyle y = 24  x\)
46.
\(x\)  \(0\)  \(4\)  \(10\)  \(\hphantom{000}\)  \(14\)  \(\hphantom{000}\) 
\(y\)  \(\hphantom{000}\)  \(\hphantom{000}\)  \(\hphantom{000}\)  \(18\)  \(\hphantom{000}\)  \(24\) 
\(x\) \(0\) \(4\) \(10\) \(12\) \(14\) \(16\) \(y\) \(0\) \(6\) \(15\) \(18\) \(21\) \(24\) \(\displaystyle y = \dfrac{3}{2}x\)
47.
\(x\)  \(0\)  \(\hphantom{000}\)  \(4\)  \(\hphantom{000}\)  \(16\)  \(25\) 
\(y\)  \(\hphantom{000}\)  \(1\)  \(\hphantom{000}\)  \(3\)  \(\hphantom{000}\)  \(\hphantom{000}\) 
\(x\) \(0\) \(1\) \(4\) \(9\) \(16\) \(25\) \(y\) \(0\) \(1\) \(2\) \(3\) \(4\) \(5\) \(\displaystyle y = \sqrt{x} \)
48.
\(x\)  \(\hphantom{000}\)  \(0.5\)  \(1\)  \(1.5\)  \(\hphantom{000}\)  \(4\) 
\(y\)  \(4\)  \(\hphantom{000}\)  \(\hphantom{000}\)  \(\hphantom{000}\)  \(0.5\)  \(\hphantom{000}\) 
\(x\) \(0.25\) \(0.50\) \(1.00\) \(1.50\) \(2.00\) \(4.00\) \(y\) \(4.00\) \(2.00\) \(1.00\) \(0.67\) \(0.50\) \(0.25\) \(\displaystyle y = \dfrac{1}{x}\)
49.
\(x\)  \(3\)  \(2\)  \(\hphantom{000}\)  \(0\)  \(1\)  \(2\) 
\(y\)  \(\hphantom{000}\)  \(\hphantom{000}\)  \(3\)  \(\hphantom{000}\)  \(\hphantom{000}\)  \(\hphantom{000}\) 
\(x\) \(3\) \(2\) \(1\) \(0\) \(1\) \(2\) \(y\) \(5\) \(0\) \(3\) \(4\) \(3\) \(0\) \(\displaystyle y = x^24 \)
50.
\(x\)  \(3\)  \(2\)  \(\hphantom{000}\)  \(0\)  \(1\)  \(\hphantom{000}\) 
\(y\)  \(\hphantom{000}\)  \(\hphantom{000}\)  \(8\)  \(\hphantom{000}\)  \(\hphantom{000}\)  \(7\) 
\(x\) \(3\) \(2\) \(1\) \(0\) \(1\) \(4\) \(y\) \(0\) \(5\) \(8\) \(9\) \(8\) \(7\) \(\displaystyle y = 9  x^2\)