Section 2.2 Some Basic Functions
ΒΆSubsection Absolute Value
The absolute value is used to discuss problems involving distance. For example, consider the number line shown below. Starting at the origin, we travel in opposite directions to reach the two numbers 6 and β6, but the distance we travel in each case is the same.Absolute Value.
The absolute value of x is defined by
Note 2.14.
This definition says that the absolute value of a positive number (or zero) is the same as the number. To find the absolute value of a negative number, we take the opposite of the number, which results in a positive number. For instance,
Example 2.15.
Simplify each expression.
\(\abs{3 - 8}\)
\(\abs{3} - \abs{8}\)
-
We simplify the expression inside the absolute value bars first.
\begin{equation*} \abs{3 - 8} = \abs{-5} = 5 \end{equation*} -
We simplify each absolute value; then subtract.
\begin{equation*} \abs{3} - \abs{8} = 3 - 8 = -5 \end{equation*}
Checkpoint 2.16.
Simplify each expression.
\(12 - 3\abs{-6}\)
\(-7 - 3\abs{2 - 9}\)
\(-6\)
\(-28\)
Subsection Examples of Models
Many situations can be modeled by a handful of simple functions. The following examples represent applications of eight useful functions.The contractor for a new hotel is estimating the cost of the marble tile for a circular lobby. The cost is a function of the square of the diameter of the lobby.
The number of board-feet that can be cut from a Ponderosa pine is a function of the cube of the circumference of the tree at a standard height.
The manager of an appliance store must decide how many coffee-makers to order every quarter. The optimal order size is a function of the square root of the annual demand for coffeemakers.
Investors are deciding whether to support a windmill farm. The wind speed needed to generate a given amount of power is a function of the cube root of the power.
The frequency of the note produced by a violin string is a function of the reciprocal of the length of the string.
The loudness, or intensity, of the music at a concert is a function of the reciprocal of the square of your distance from the speakers.
The annual return on an investment is a linear function of the interest rate.
You are flying from Los Angeles to New York. Your distance from the Mississippi River is an absolute value function of time.
Investigation 2.3. Eight Basic Functions.
Part I Some Powers
- Complete the table of values for the squaring function, f(x)=x2, and the cubing function, g(x)=x3. Then sketch each function on graph paper, using the table values to help you scale the axes.
- Verify both graphs with your graphing calculator.
- State the intervals on which each graph is increasing.
- Write a few sentences comparing the two graphs. The graph of y=x2 is called a parabola, and the graph of y=x3 is called a cubic.
x | f(x)=x2 | g(x)=x3 |
β3 | ||
β2 | ||
β1 | ||
β12 | ||
0 | ||
12 | ||
1 | ||
2 | ||
3 |
Part II Some Roots
- Complete the tables for the square root function, f(x)=βx, and the cube root function, g(x)=3βx. (Round your answers to two decimal places.) Then sketch each function on graph paper, using the table values to help you scale the axes.
- Verify both graphs with your graphing calculator.
- State the intervals on which each graph is increasing.
- Write a few sentences comparing the two graphs.
x | f(x)=βx |
0 | |
12 | |
1 | |
2 | |
3 | |
4 | |
5 | |
7 | |
9 |
x | g(x)=3βx |
β8 | |
β4 | |
β1 | |
β12 | |
0 | |
12 | |
1 | |
4 | |
8 |
Part III Asymptotes
- Complete the table for the functionsf(x)=1x and g(x)=1x2What is true about f(0) and g(0)?
- Prepare a grid on graph paper, scaling both axes from β5 to 5. Plot the points from the table and connect them with smooth curves.
- As x increases through larger and larger values, what happens to the values of f(x)? Extend your graph to reflect your answer.
- What happens to f(x) as x decreases through larger and larger negative values (that is, for x=β5,β6,β7,β¦)? Extend your graph for these x-values.
x | f(x)=1x | g(x)=1x2 |
β4 | ||
β3 | ||
β2 | ||
β1 | ||
β12 | ||
0 | ||
12 | ||
1 | ||
2 | ||
3 | ||
4 |
As the values of x get larger in absolute value, the graph approaches the x-axis. However, because 1x never equals zero for any x-value, the graph never actually touches the x-axis. We say that the x-axis is a horizontal asymptote for the graph.
Repeat step (3) for the graph of g(x).
Next we'll examine the graphs of f and g near x=0.
-
Use your calculator to evaluate f for several x-values close to zero and record the results in the tables below.
x f(x)=1x g(x)=1x2 β2 β1 β0.1 β0.01 β0.001 x f(x)=1x g(x)=1x2 2 1 0.1 0.01 0.001 What happens to the values of f(x) as x approaches zero? Extend your graph of f to reflect your answer.
As x approaches zero from the left (through negative values), the function values decrease toward ββ. As x approaches zero from the right (through positive values), the function values increase toward β. The graph approaches but never touches the vertical line x=0 (the y-axis.)We say that the graph of f has a vertical asymptote at x=0.
- Repeat step (1) for the graph of g(x).
- The functions f(x)=1x and g(x)=1x2 are examples of rational functions, so called because they are fractions, or ratios. Verify both graphs with your graphing calculator. Use the windowXmin=β4Xmax=4Ymin=β4Ymax=4
- State the intervals on which each graph is increasing.
- Write a few sentences comparing the two graphs.
Part IV Absolute Value
- Complete the table for the two functions f(x)=x and g(x)=|x|. Then sketch each function on graph paper, using the table values to help you scale the axes.
- Verify both graphs with your graphing calculator. Your calculator uses the notation abs(x) instead of |x| for the absolute value of x. First, position the cursor after Y1= in the graphing window. Now access the absolute value function by pressing
2nd
0 for CATALOG; thenENTER
for abs(. Donβt forget to press X if you want to graph y=|x|. - State the intervals on which each graph is increasing.
- Write a few sentences comparing the two graphs.
x | f(x)=x | g(x)=|x| |
β4 | ||
β3 | ||
β2 | ||
β1 | ||
β12 | ||
0 | ||
12 | ||
1 | ||
2 | ||
3 | ||
4 |
Subsection Graphs of Eight Basic Functions
The graphs of the eight basic functions considered in Investigation 2.3 are shown below. Once you know the shape of each graph, you can sketch an accurate picture by plotting a few guidepoints and drawing the curve through those points. Usually, points (or vertical asymptotes!) at x=β1, 0, and 1 make good guidepoints.Subsection Properties of the Basic Functions
In Section 1.2, we saw that for most functions, f(a+b) is not equal to f(a)+f(b). We may be able to find some values of a and b for which f(a+b)=f(a)+f(b) is true, but if it is not true for all values of a and b, we cannot claim that f(a+b)=f(a)+f(b) for that function. For example, for the function f(x)=x2, if we choose a=3 and b=4, thenProperties of Absolute Value.
Example 2.17.
Verify the triangle inequality for three cases: \(a\) and \(b\) are both positive, \(a\) and \(b\) are both negative, and \(a\) and \(b\) have opposite signs.
-
We choose positive values for \(a\) and \(b\text{,}\) say \(a = 3\) and \(b = 5\text{.}\) Then
\begin{equation*} \abs{3 + 5} = \abs{8} = 8 ~\text{ and } ~ \abs{3}+\abs{5}= 3 + 5 = 8 \end{equation*}so \(\abs{3 + 5} = \abs{3} + \abs{5}\text{.}\)
-
For the second case, we choose \(a = -3\) and \(b = -5\text{.}\) Then
\begin{equation*} \abs{-3 + (-5)} = \abs{-8} = 8 ~\text{ and } ~ \abs{-3}+\abs{-5} = 3 + 5 = 8 \end{equation*}so \(\abs{-3 + (-5)} =\abs{-3}+\abs{-5}\text{.}\)
-
For the third case, we choose \(a = 3\) and \(b = -5\text{.}\) Then
\begin{equation*} \abs{3 + (-5)}=\abs{-2} = 2 ~\text{ and } ~ \abs{3}+\abs{-5}=3 + 5 = 8 \end{equation*}so \(\abs{3 + (-5)} \lt \abs{3}+\abs{-5}\text{.}\) In each case, \(\abs{a + b}\le\abs{a} +\abs{b}\text{.}\)
Note 2.18.
Note that verifying a statement for one or two values of the variables does not prove the statement is true for all values of the variables. However, working with examples can help us understand the meaning and significance of mathematical properties.
Checkpoint 2.19.
Verify the multiplicative property of absolute value for the three cases in Example 2.17.
- \(\displaystyle \abs{3}\abs{5}=15=\abs{3\cdot 5} \)
- \(\displaystyle \abs{-3}\abs{-5}=15=\abs{(-3)\cdot (-5)}\)
- \(\displaystyle \abs{3}\abs{-5}=15=\abs{3(-5)}\)
Subsection Functions Defined Piecewise
A function may be defined by different formulas on different portions of the x-axis. Such a function is said to be defined piecewise. To graph a function defined piecewise, we consider each piece of the x-axis separately.Example 2.20.
Graph the function defined by
Think of the plane as divided into two regions by the vertical line \(x = 1\text{,}\) as shown below. In the left-hand region (\(x \le 1\)), we graph the line \(y = x + 1\text{.}\) (The fastest way to graph the line is to plot its intercepts, \((-1, 0)\) and \((0, 1)\text{.}\))
Notice that the value \(x = 1\) is included in the first region, so \(f (1) = 1 + 1 = 2\text{,}\) and the point \((1, 2)\) is included on the graph. We indicate this with a solid dot at the point \((1, 2)\text{.}\)
In the right-hand region (\(x \gt 1\)), we graph the horizontal line \(y = 3\text{.}\) The value \(x = 1\) is not included in the second region, so the point \((1, 3)\) is not part of the graph. We indicate this with an open circle at the point \((1, 3)\text{.}\)
Checkpoint 2.21.
Graph the piecewise defined function
Example 2.22.
Write a piecewise definition for \(g(x) = \abs{x - 3}\text{.}\)
Sketch a graph of \(g(x) = \abs{x - 3}\text{.}\)
-
In the definition for \(\abs{x}\text{,}\) we replace \(x\) by \(x - 3\) to get
\begin{equation*} g(x) =\abs{x-3} = \begin{cases} x-3 \amp \text{if }~ x-3\ge 0\\ -(x-3) \amp \text{if }~ x-3\lt 0 \end{cases} \end{equation*}We can simplify this expression to
\begin{equation*} g(x) =\abs{x-3} = \begin{cases} x-3 \amp \text{if }~ x\ge 3\\ -x+3 \amp \text{if }~ x\lt 3 \end{cases} \end{equation*} -
In the first region, \(x \ge 3\text{,}\) we graph the line \(y = x - 3\text{.}\) Because \(x = 3\) is included in this region, the endpoint of this portion of the graph, \((3, 0)\text{,}\) is included, too.
In the second region, \(x \lt 3\text{,}\) we graph the line \(y = -x + 3\text{.}\) Note that the two pieces of the graph meet at the point \((0, 3)\text{,}\) as shown below.
Checkpoint 2.23.
Use your calculator to graph \(g(x) =\abs{x-3}\) and \(h(x) =\abs{x} + \abs{-3}\text{.}\) Are the graphs the same?
Explain why the functions \(f (x) =\abs{x + k}\) and \(g(x) =\abs{x} + \abs{k}\) are not the same if \(k\ne 0\text{.}\)
No
Because \(\abs{x + k}\ne \abs{x} + \abs{k}\) when \(x\) and \(k\) have opposite signs.
Subsection Section Summary
ΒΆSubsubsection Vocabulary
Look up the definitions of new terms in the Glossary.Absolute value
Verify
Triangle inequality
Vertical asymptote
Rational function
Parabola
Piecewise defined function
Multiplicative property
Guidepoints
Horizontal asymptote
Cubic
Subsubsection CONCEPTS
-
The absolute value of x is defined by
|x|={xif xβ₯0βxif x<0 -
The absolute value has the following properties:
|a+b|β€|a|+|b| Triangle inequality |ab|=|a||b| Multiplicative property Many useful functions fall into families or classes of variations on basic functions.
We can make sketches of the eight basic functions using guidepoints.
Functions can be defined piecewise, with different formulas on different intervals
Subsubsection STUDY QUESTIONS
Is it true that βx must be a negative number? Why or why not?
Are there any numbers for which x=βx?
If 0<x<1, which is larger, x2 or x3?
If 0<x<1, βx or 3βx?
List the eight basic functions considered in this section.
Which of the eight basic functions have a horizontal asymptote? A vertical asymptote?
What does an open circle on a graph mean?
For what value(s) of x does |x+6|=0?
Subsubsection SKILLS
Practice each skill in the Homework problems listed.Simplify expressions containing absolute values: #1β10
Sketch graphs of the basic functions by hand: #15β18
Identify the graph of a basic function: #19β26
Solve equations and inequalities graphically: #11β14, 27β34
Graph functions defined piecewise: #41β58
Exercises Homework 2.2
ΒΆFor problems 1-10, simplify the expression according to the order of operations.
In Problems 11β14, show how to use the graphs to find the values. Estimate your answers to one decimal point. Compare your estimates to values obtained with a calculator.
11.
Refer to the graph of f(x)=x3.
Estimate the value of (1.4)3.
Find all numbers whose cubes are β20.
Find all solutions of the equation x3=6.
Estimate the value of 3β24.
12.
Refer to the graph of f(x)=x2.
Estimate the value of (β2.5)2.
Find all numbers whose squares are 12.
Find all solutions of the equation x2=15.
Estimate the value of β10.5.
13.
Refer to the graph of f(x)=1x.
Estimate the value of 13.4.
Find all numbers whose reciprocals are β2.5.
Find all solutions of the equation 1x=4.8.
14.
Refer to the graph of f(x)=|xβ2|.
Estimate the value of |1.6β2|.
Find all values of x for which |xβ2|=3.
Find all solutions of the equation |xβ2|=0.4.
For Problems 15β18,
Sketch both functions on the same grid, paying attention to the shape of the graph. Plot at least three guidepoints for each graph to ensure accuracy.
Use the graph to find all solutions of the equation f(x)=g(x).
On what intervals is f(x)>g(x)?
For Problems 19-24, graph each set of functions together in the ZDecimal window. Describe how graphs (b) and (c) are different from the basic graph.
Each graph in Problems 25-26 is a variation of one of the eight basic graphs of Investigation 2.3. Identify the basic graph for each problem.
In Problems 27β30, use the graph to estimate the solution to the equation or inequality. Show the solution or solutions on the graph.
27.
The figure shows a graph of f(x)=βxβ2, for x>0. Solve the following:
βxβ2=1.5
βxβ2=2.25
βxβ2<1
βxβ2>β0.25
28.
The figure shows a graph of g(x)=4x+2, for x>β2. Solve the following:
4x+2=4
4x+2=0.8
4x+2>1
4x+2<3
29.
The figure shows a graph of w(t)=β10(t+1)3+10. Solve the following:
β10(t+1)3+10=100
β10(t+1)3+10=β140
β10(t+1)3+10>β50
β20<β10(t+1)3+10<40
30.
The figure shows a graph of H(z)=43βzβ4+6. Solve the following:
43βzβ4+6=2
43βzβ4+6=12
43βzβ4+6>14
43βzβ4+6<6
For Problems 31-34, gaph the function with the ZInteger setting. Use the graph to solve each equation or inequality. Check your solutionsalgebraically.
31.
Graph F(x)=4βxβ25.
Solve ~~4\sqrt{x - 25}=16
Solve ~~8\lt 4\sqrt{x - 25}\le 24
32.
Graph ~~G(x) = 15 - 0.01(x - 2)^3 \text{.}
Solve ~~15 - 0.01(x - 2)^3 =-18.75
Solve ~~15 - 0.01(x - 2)^3 \le 25
33.
Graph ~~H(x) = 24 - 0.25(x - 6)^2\text{.}
Solve ~~24 - 0.25(x - 6)^2=-6.25
Solve ~~24 - 0.25(x - 6)^2\gt 11.75
34.
Graph ~~R(x) = 0.1(x + 12)^2 - 18 \text{.}
Solve ~~0.1(x + 12)^2 - 18 =14.4
Solve ~~0.1(x + 12)^2 - 18 \lt 4.5
For Problems 35β40,
Graph the equation by completing the table and plotting points.
Does the equation define y as a function of x\text{?} Why or why not?
35.
x=y^2
x | \hphantom{000} | \hphantom{000} | \hphantom{000} | \hphantom{000} | \hphantom{000} | \hphantom{000} | \hphantom{000} |
y | -2 | -1 | -\frac{1}{2} | 0 | \frac{1}{2} | 1 | 2 |
36.
x=y^3
x | \hphantom{000} | \hphantom{000} | \hphantom{000} | \hphantom{000} | \hphantom{000} | \hphantom{000} | \hphantom{000} |
y | -2 | -1 | -\frac{1}{2} | 0 | \frac{1}{2} | 1 | 2 |
37.
x=\abs{y}
x | \hphantom{000} | \hphantom{000} | \hphantom{000} | \hphantom{000} | \hphantom{000} | \hphantom{000} | \hphantom{000} |
y | -2 | -1 | -\frac{1}{2} | 0 | \frac{1}{2} | 1 | 2 |
38.
\abs{x}=\abs{y}
x | \hphantom{000} | \hphantom{000} | \hphantom{000} | \hphantom{000} | \hphantom{000} | \hphantom{000} | \hphantom{000} |
y | -2 | -1 | -\frac{1}{2} | 0 | \frac{1}{2} | 1 | 2 |
39.
x=\dfrac{1}{y}
x | \hphantom{000} | \hphantom{000} | \hphantom{000} | \hphantom{000} | \hphantom{000} | \hphantom{000} | \hphantom{000} |
y | -2 | -1 | -\frac{1}{2} | 0 | \frac{1}{2} | 1 | 2 |
40.
x=\dfrac{1}{y^2}
x | \hphantom{000} | \hphantom{000} | \hphantom{000} | \hphantom{000} | \hphantom{000} | \hphantom{000} | \hphantom{000} |
y | -2 | -1 | -\frac{1}{2} | 0 | \frac{1}{2} | 1 | 2 |
For Problems 41β52, graph the following piecewise defined functions. Indicate whether the endpoints of each piece are included on the graph.
41.
f(x) = \begin{cases} -2 \amp \text{if } x\le 1\\ x-3 \amp \text{if } x\gt 1 \end{cases}
42.
h(x) = \begin{cases} -x+2 \amp \text{if } x\le -1\\ 3 \amp \text{if } x\gt -1 \end{cases}
43.
G(t) = \begin{cases} 3t+9 \amp \text{if } t\lt -2\\ -3-\dfrac{1}{2}t \amp \text{if } t\ge -2 \end{cases}
44.
F(s) = \begin{cases} \dfrac{1}{3}s+3 \amp \text{if } s\lt 3\\ 2s-3 \amp \text{if } s\ge 3 \end{cases}
45.
H(t) = \begin{cases} t^2 \amp \text{if } t\le 1\\ \dfrac{1}{2}t+\dfrac{1}{2} \amp \text{if } t\gt 1 \end{cases}
46.
g(t) = \begin{cases} \dfrac{3}{2}t+7 \amp \text{if } t\le -2\\ t^2 \amp \text{if } t\gt -2 \end{cases}
47.
k(x) = \begin{cases} \abs{x} \amp \text{if } x\le 2\\ \sqrt{x} \amp \text{if } x\gt 2 \end{cases}
48.
S(x) = \begin{cases} \dfrac{1}{x} \amp \text{if } x\lt 1\\ \abs{x} \amp \text{if } x\ge 1 \end{cases}
49.
D(x) = \begin{cases} \abs{x} \amp \text{if } x\lt -1\\ x^3 \amp \text{if } x\ge -1 \end{cases}
50.
m(x) = \begin{cases} x^2 \amp \text{if } x\le \dfrac{1}{2} \\ \abs{x} \amp \text{if } x\gt \dfrac{1}{2} \end{cases}
51.
P(t) = \begin{cases} t^3 \amp \text{if } t\le 1\\ \dfrac{1}{t^2} \amp \text{if } t\gt 1 \end{cases}
52.
Q(t) = \begin{cases} t^2 \amp \text{if } t\le -1\\ \sqrt[3]{t} \amp \text{if } t\gt -1 \end{cases}
For Problems 53-58, write a piecewise definition for the function and sketch its graph.
In Problems 59β64, decide whether each statement is true for all values of a and b\text{.} If the statement is true, give an algebraic justification. If it is false, find values of a and b to disprove it.
f(a + b) = f(a) + f(b)
f(ab) = f(a)f(b)
65.
Verify that \abs{a - b} gives the distance between a and b on a number line.
a=3, ~b=8
a=-2, ~b=-6
a=4, ~b=-3
a=-2, ~b=5
66.
Which of the following statements is true for all values of a and b\text{?}
\abs{a-b}=\abs{a}+\abs{b}
\abs{a-b}\le \abs{a}+\abs{b}
\abs{a-b}\ge \abs{a}+\abs{b}
67.
Explain how the distributive law, a(b + c) = ab + ac\text{,} is different from the equation f (a + b) = f (a) + f (b) .
68.
For each function, decide whether f (kx) = k f (x) for all x\ne 0\text{,} where k\ne 0 is a constant.
f(x)=x^2
f(x)=\dfrac{1}{x}
f(x)=\sqrt{x}
f(x)=\abs{x}
For Problems 69-70,find the indicated value.
69.
Use the function F(s) = \begin{cases} \dfrac{1}{3}s+3 \amp \text{if } s\lt 3\\ 2s-3 \amp \text{if } s\ge 3 \end{cases} (from Problem 44) and add the indicated values
F(0)
F(3)
F(β9)
70.
Use the function k(x) = \begin{cases} \abs{x} \amp \text{if } x\le 2\\ \sqrt{x} \amp \text{if } x\gt 2 \end{cases} (from Problem 47) and add the indicated values
k(2)
k(1)
k(-3)
k(4.5)