section-2-2.xml

<?xml version="1.0" encoding="UTF-8" ?>
<!-- <mathbook><book> -->

<section xml:id="basic-functions"   xmlns:xi="http://www.w3.org/2001/XInclude">
<title>Some Basic Functions</title><introduction>
<p>
    In this section, we study the graphs of some important basic functions. Many functions fall into families or classes of similar functions, and recognizing the appropriate family for a given situation is an important part of modeling.
</p>
<p>
    We begin by reviewing the absolute value.
</p>
</introduction>
<subsection><title>Absolute Value</title>  
<p>
    The absolute value is used to discuss problems involving distance. For example, consider the number line in <xref ref="fig-abs-six" text="type-global" />. Starting at the origin, we travel in opposite directions to reach the two numbers <m>6</m> and <m>-6</m>, but the distance we travel in each case is the same.
</p> 
<figure xml:id="fig-abs-six"><caption></caption><image source="images/fig-abs-six"  width="60%"><description>absolute value</description></image></figure>
<p>
    The distance from a number <m>c</m> to the origin is called the <term>absolute value</term><idx>absolute value</idx> of <m>c</m>, denoted by <m>\abs{c}</m>. Because distance is never negative, the absolute value of a number is always positive (or zero). Thus, <m>\abs{6}= 6</m> and <m>\abs{-6} = 6</m>. In general, we define the absolute value of a number <m>x</m> as follows.
</p>

<assemblage><title>Absolute Value</title>
<p>
   The absolute value of <m>x</m> is defined by
   <me>
    \abs{x} =
    \begin{cases}
    x \amp \text{if } x\ge 0\\
    -x  \amp \text{if } x\lt 0
    \end{cases}
    </me>
</p>
</assemblage>

<note><p>
    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,
    <me>\abs{-6} = -(-6) = 6</me></p></note>
    <p>
    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.
</p>

<example xml:id="example-abs">
    <p>Simplify each expression.</p>
    <ol label="a">
        <li><p><m>\abs{3 - 8}</m></p></li>
        <li><p><m>\abs{3} - \abs{8}</m></p></li>
    </ol>

<solution>
    <ol label="a">
        <li><p>Simplify the expression inside the absolute value bars first.
            <me>\abs{3 - 8} = \abs{-5} = 5</me></p></li>
        <li><p>Simplify each absolute value; then subtract.
            <me>\abs{3} - \abs{8} = 3 - 8 = -5</me></p></li>
    </ol>
</solution>
</example>

<exercise xml:id="exercise-abs"><statement>
    Simplify each expression.
    <ol label="a">
        <li><p><m>12 - 3\abs{-6}</m></p></li>
        <li><p><m>-7 - 3\abs{2 - 9}</m></p></li>
    </ol>
</statement>
<answer><p>
    <ol label="a" cols="2">
        <li><p><m>-6</m></p></li>
        <li><p><m>-28</m></p></li>
    </ol>
</p></answer>
</exercise>
</subsection>


<subsection><title>Examples of Models</title>
    <p>
        Many situations can be modeled by a handful of simple functions. The following examples represent applications of eight useful functions.
    </p>



<sidebyside widths="45% 45%"><p>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 <em>square</em> of the diameter of the lobby.</p>
<p>The number of board-feet that can be cut from a Ponderosa pine is a function of the <em>cube</em> of the circumference of the tree at a standard height.</p></sidebyside>

<sidebyside widths="30% 30%">
    <figure xml:id="fig-quadratic-model"><caption></caption><image source="images/fig-quadratic-model"><description>parabola</description></image></figure>

    <figure xml:id="fig-cubic-model"><caption></caption><image source="images/fig-cubic-model"><description>cubic</description></image></figure>
</sidebyside>


<sidebyside widths="45% 45%"><p>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 <em>square root</em> of the annual demand for coffeemakers.</p>
<p>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 <em>cube root</em> of the power.</p></sidebyside>

<sidebyside widths="30% 30%">
    <figure xml:id="fig-sqrt-model"><caption></caption><image source="images/fig-sqrt-model"><description>square root</description></image></figure>

    <figure xml:id="fig-cube-root-model"><caption></caption><image source="images/fig-cube-root-model"><description>cube root</description></image></figure>
</sidebyside>


<sidebyside widths="45% 45%"><p>The frequency of the note produced by a violin string is a function of the <em>reciprocal</em> of the length of the string.</p>
<p>The loudness, or intensity, of the music at a concert is a function of the <em>reciprocal of the square</em> of your distance from the speakers.</p></sidebyside>

<sidebyside widths="30% 30%">
    <figure xml:id="fig-inverse-model"><caption></caption><image source="images/fig-inverse-model"><description>inverse</description></image></figure>

    <figure xml:id="fig-inverse-square-model"><caption></caption><image source="images/fig-inverse-square-model"  width="60%"><description>cube root</description></image></figure>
</sidebyside>



<sidebyside widths="45% 45%"><p>The annual return on an investment is a linear function of the interest rate.</p>
<p>You are flying from Los Angeles to New York. Your distance from the Mississippi River is an <em>absolute value</em> function of time.</p></sidebyside>

<sidebyside widths="30% 30%">
    <figure xml:id="fig-linear-model"><caption></caption><image source="images/fig-linear-model"><description>linear</description></image></figure>

    <figure xml:id="fig-abs-model"><caption></caption><image source="images/fig-abs-model"><description>absolute value</description></image></figure>
</sidebyside>


<p>
    We will consider each of these functions and their applications in more detail in later sections. For now, you should become familiar with the properties of each graph and be able to sketch them easily from memory.
</p>

<investigation xml:id="Eight-basic-functions"><title>Eight Basic Functions</title>
<statement>
<p>Part I Some Powers and Roots</p>

<ol>
    <li><p>Complete <xref ref="table-quadratic-cubic" text="type-global"/>, a table of values for the squaring function, <m>f (x) = x^2</m>, and the cubing function, <m>g(x) = x^3</m>. Then sketch each function on graph paper, using the table values to help you scale the axes.</p></li>
    <li><p>Verify both graphs with your graphing calculator.</p></li>
    <li><p>State the intervals on which each graph is increasing.</p></li>
    <li><p>Write a few sentences comparing the two graphs. The graph of <m>y = x^2</m> is called a <term>parabola</term><idx>parabola</idx>, and the graph of <m>y = x^3</m> is called a <term>cubic</term><idx>cubic</idx>.</p></li>
    <li><p>Complete <xref ref="table-sqrt" text="type-global"/> and <xref ref="table-cube-root" text="type-global"/> for the square root function, <m>f (x) = \sqrt{x}</m>, and the cube root function, <m>g(x) = \sqrt[3]{x} </m>. (Round your answers to two decimal places.) Then sketch each function on graph paper, using the table values to help you scale the axes.</p>
    <p></p>

<sidebyside>
<table xml:id="table-quadratic-cubic"><tabular top="major" halign="center" right="minor" left="minor" bottom="minor">
        <row bottom="minor">
            <cell><m>x</m></cell>
            <cell><m>f(x)=x^2</m></cell>
            <cell><m>g(x)=x^3</m></cell>
        </row>
        <row>
            <cell><m>-3</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>-2</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>-1</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>-\frac{1}{2}</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>0</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>\frac{1}{2}</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>1</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>2</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>3</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
    </tabular><caption></caption></table>

<table xml:id="table-sqrt"><tabular top="major" halign="center" right="minor" left="minor" bottom="minor">
        <row bottom="minor">
            <cell><m>x</m></cell>
            <cell><m>f(x)=\sqrt{x}</m></cell>
        </row>
        <row>
            <cell><m>0</m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>\frac{1}{2}</m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>1</m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>2</m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>3</m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>4</m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>5</m></cell>
            <cell><m></m></cell>
        </row>
       <row>
            <cell><m>7</m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>9</m></cell>
            <cell><m></m></cell>
        </row>
     </tabular><caption></caption></table>

     <table xml:id="table-cube-root"><tabular top="major" halign="center" right="minor" left="minor" bottom="minor">
        <row bottom="minor">
            <cell><m>x</m></cell>
            <cell><m>g(x)=\sqrt[3]{x}</m></cell>
        </row>
        <row>
            <cell><m>-8</m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>-4</m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>-1</m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>-\frac{1}{2}</m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>0</m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>\frac{1}{2}</m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>1</m></cell>
            <cell><m></m></cell>
        </row>
       <row>
            <cell><m>4</m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>8</m></cell>
            <cell><m></m></cell>
        </row>
    </tabular><caption></caption></table>
</sidebyside>
</li>
    <li><p>Verify both graphs with your graphing calculator.</p></li>
    <li><p>State the intervals on which each graph is increasing.</p></li>
    <li><p>Write a few sentences comparing the two graphs.</p></li>
</ol>

<p>Part II Asymptotes</p>

<ol>
    <li><p>Complete <xref ref="table-reciprocals" text="type-global"/> for the functions
        <me>f(x)=\dfrac{1}{x} ~\text{ and } ~ g(x)=\dfrac{1}{x^2}</me>
    What is true about <m>f (0)</m> and <m>g(0)</m>?

<table xml:id="table-reciprocals"><tabular top="major" halign="center" right="minor" left="minor" bottom="minor">
        <row bottom="minor">
            <cell><m>x</m></cell>
            <cell><m>f(x)=\dfrac{1}{x}</m></cell>
            <cell><m>g(x)=\dfrac{1}{x^2}</m></cell>
        </row>
        <row>
            <cell><m>-4</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>-3</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>-2</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>-1</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>-\frac{1}{2}</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>0</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>\frac{1}{2}</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>1</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>2</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>3</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>4</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
    </tabular><caption></caption></table>

    </p></li>
    <li><p>
        Prepare a grid on graph paper, scaling both axes from <m>-5</m> to <m>5</m>. Plot the points from <xref ref="table-reciprocals" text="type-global"/> and connect them with smooth curves.
    </p></li>
    <li>
        <p>As <m>x</m> increases through larger and larger values, what happens to the values of <m>f (x)</m>? Extend your graph to reflect your answer. </p> 
        <p>What happens to <m>f (x)</m> as <m>x</m> decreases through larger and larger negative values (that is, for <m>x = -5,-6,-7, \ldots </m>)? Extend your graph for these <m>x</m>-values. </p> 
        <p>As the values of <m>x</m> get larger in absolute value, the graph approaches the <m>x</m>-axis. However, because <m>\dfrac{1}{x}</m> never <em>equals</em> zero for any <m>x</m>-value, the graph never actually touches the <m>x</m>-axis. We say that the <m>x</m>-axis is a <term>horizontal asymptote</term><idx>horizontal asymptote</idx> for the graph.</p>
    </li>
    <li><p>Repeat step (3) for the graph of <m>g(x)</m>.</p></li>
    <li><p>
        Next we will examine the graphs of <m>f</m> and <m>g</m> near <m>x = 0</m>. Use your calculator to evaluate <m>f</m> for several <m>x</m>-values close to zero and record the results in <xref ref="table-reciprocals2a" text="type-global"/> and <xref ref="table-reciprocals2b" text="type-global"/>.
        <p></p>
    <sidebyside><table xml:id="table-reciprocals2a"><tabular top="major" halign="center" right="minor" left="minor" bottom="minor">
        <row bottom="minor">
            <cell><m>x</m></cell>
            <cell><m>f(x)=\dfrac{1}{x}</m></cell>
            <cell><m>g(x)=\dfrac{1}{x^2}</m></cell>
        </row>
        <row>
            <cell><m>-2</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>-1</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>-0.1</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>-0.01</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>-0.001</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
    </tabular><caption></caption></table>
    <table xml:id="table-reciprocals2b"><tabular top="major" halign="center" right="minor" left="minor" bottom="minor">
        <row bottom="minor">
            <cell><m>x</m></cell>
            <cell><m>f(x)=\dfrac{1}{x}</m></cell>
            <cell><m>g(x)=\dfrac{1}{x^2}</m></cell>
        </row>
        <row>
            <cell><m>2</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>1</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>0.1</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>0.01</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>0.001</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
    </tabular><caption></caption></table></sidebyside>
    <p>What happens to the values of <m>f (x)</m> as <m>x</m> approaches zero? Extend your graph of <m>f</m> to reflect your answer.</p>
    <p>As <m>x</m> approaches zero from the left (through negative values), the function values decrease toward <m>-\infty</m>. As <m>x</m> approaches zero from the right (through positive values), the function values increase toward <m>\infty</m>. The graph approaches but never touches the vertical line <m>x = 0</m> (the <m>y</m>-axis.)We say that the graph of <m>f</m> has a <term>vertical asymptote</term><idx>vertical asymptote</idx> at <m>x = 0</m>.</p>
    </p></li>
    <li><p>
        Repeat step (5) for the graph of <m>g(x)</m>.
    </p></li>
    <li><p>
        The functions <m>f (x) = \dfrac{1}{x}</m> and <m>g(x) = \dfrac{1}{x^2}</m> are examples of <term>rational functions</term><idx>rational functions</idx>, so called because they are fractions, or ratios. Verify both graphs with your graphing calculator. Use the window
        \begin{align}
        \text{Xmin} \amp = -4 \amp\amp \text{Xmax} = 4\\
        \text{Ymin} \amp = -4 \amp\amp \text{Ymax} = 4
        \end{align}
    </p></li>
    <li><p>
        State the intervals on which each graph is increasing.
    </p></li>
    <li><p>
        Write a few sentences comparing the two graphs.
    </p></li>
</ol>

<p>Part III Absolute Value</p>
<ol>
    <li><p>
        Complete <xref ref="table-linear-abs" text="type-global"/> for the two functions <m>f (x) = x</m> and <m>g(x) = \abs{x}</m>. Then sketch each function on graph paper, using the table values to help you scale the axes.
            <table xml:id="table-linear-abs"><tabular top="major" halign="center" right="minor" left="minor" bottom="minor">
        <row bottom="minor">
            <cell><m>x</m></cell>
            <cell><m>f(x)=x</m></cell>
            <cell><m>g(x)=\abs{x}</m></cell>
        </row>
        <row>
            <cell><m>-4</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>-3</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>-2</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>-1</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>-\frac{1}{2}</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>0</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>\frac{1}{2}</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>1</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>2</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>3</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell><m>4</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
     </tabular><caption></caption></table>
    </p></li>
    <li><p>
        Verify both graphs with your graphing calculator. Your calculator uses the notation <m>abs (x)</m> instead of <m>\abs{x}</m> for the absolute value of <m>x</m>. First, position the cursor after <m>Y_1 =</m> in the graphing window. Now access the absolute value function by pressing <c>2nd</c> <m>0</m> for <em>CATALOG</em>; then <c>ENTER</c> for <em>abs(</em>. Don’t forget to press <m>X</m> if you want to graph <m>y = \abs{x}</m>.
    </p></li>
    <li><p>
        State the intervals on which each graph is increasing.
    </p></li>
    <li><p>
        Write a few sentences comparing the two graphs.
    </p></li>
</ol></statement>
</investigation>
</subsection>

<subsection><title>Graphs of Eight Basic Functions</title>
    <p>
        The graphs of the eight basic functions considered in <xref ref="Eight-basic-functions" text="type-global"/> 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 <m>x = -1</m>, <m>0</m>, and <m>1</m> make good guidepoints.
    </p>
<figure xml:id="fig-basic-a-c"><caption></caption><image source="images/fig-basic-a-c"  width="100%"><description>parabola, cubic, square root</description></image></figure>
<figure xml:id="fig-basic-d-f"><caption></caption><image source="images/fig-basic-d-f"  width="100%"><description>parabola, cubic, square root</description></image></figure>
<figure xml:id="fig-basic-g-h"><caption></caption><image source="images/fig-basic-g-h"  width="60%"><description>parabola, cubic, square root</description></image></figure>

</subsection>

<subsection><title>Properties of the Basic Functions</title>
<p>
    In <xref ref="functions" text="type-global"/>, we saw that for most functions, <m>f (a + b)</m> is not equal to <m>f (a) + f (b)</m>. We may be able to find <em>some</em> values of <m>a</m> and <m>b</m> for which <m>f (a + b) = f (a) + f (b)</m> is true, but if it is not true for <em>all</em> values of <m>a</m> and <m>b</m>, we cannot claim that <m>f (a + b) = f (a) + f (b)</m> for that function.</p> 
    <p>For example, for the function <m>f (x) = x^2</m>, if we choose <m>a = 3</m> and <m>b = 4</m>, then
    <md><mrow>f (3 + 4) \amp = f (7) = 7^2 = 49</mrow>
        <mrow>\text{but }~ f (3) + f (4) \amp = 3^2 + 4^2 = 9 + 16 = 25</mrow>
    </md>
    so we have proved that <m>f (a + b) \ne f (a) + f (b)</m> for the squaring function. (In fact, we already knew this because <m>(a + b)^2 \ne a^2 + b^2</m> as long as neither <m>a</m> nor <m>b</m> is <m>0</m>.)
</p>
<p>
    What about multiplication? Which of the basic functions have the property that <m>f (ab) = f (a) f (b)</m> for all <m>a</m> and <m>b</m>? You will consider this question in the homework problems, but in particular you will need to recall the following  properties of absolute
value.
</p>

<assemblage><title>Properties of Absolute Value</title>
<p>\begin{align}
\abs{a + b} \le \abs{a} + \abs{b} \amp\amp \text{Triangle inequality}\\
\abs{a  b} = \abs{a}  \abs{b} \amp\amp \text{Multiplicative property }
\end{align}</p>   
</assemblage>

<example xml:id="example-triangle-inequality">
    <p>
        Verify the triangle inequality for three cases: <m>a</m> and <m>b</m> are both positive, <m>a</m> and <m>b</m> are both negative, and <m>a</m> and <m>b</m> have opposite signs.
    </p>
<solution><p>
    <ul>
        <li><p>
    We choose positive values for <m>a</m> and <m>b</m>, say <m>a = 3</m> and <m>b = 5</m>. Then 
    <me>\abs{3 + 5} = \abs{8} = 8 ~\text{ and } ~ \abs{3}+\abs{5}= 3 + 5 = 8</me>
    so <m>\abs{3 + 5} = \abs{3} + \abs{5}</m>.</p></li>
    <li><p> For the second case, we choose <m>a = -3</m> and <m>b = -5</m>. Then
    <me>\abs{-3 + (-5)} = \abs{-8} = 8 ~\text{ and } ~ \abs{-3}+\abs{-5} = 3 + 5 = 8</me>
    so <m>\abs{-3 + (-5)} =\abs{-3}+\abs{-5}</m>.</p></li>
    <li><p> For the third case, we choose <m>a = 3</m> and <m>b = -5</m>. Then
    <me>\abs{3 + (-5)}=\abs{-2} = 2  ~\text{ and } ~ \abs{3}+\abs{-5}=3 + 5 = 8</me>
    so <m>\abs{3 + (-5)} \lt \abs{3}+\abs{-5}</m>. In each case, <m>\abs{a + b}\le\abs{a} +\abs{b}</m>.</p></li>
</ul>
</p></solution></example>

<note><p>
    Note that <em>verifying</em> a statement for one or two values of the variables does not <em>prove</em> the statement is true for <em>all</em> values of the variables. However, working with examples can help us understand the meaning and significance of mathematical properties.
</p></note>

<exercise xml:id="exercise-abs-multiplicative-property"><statement>
    Verify the multiplicative property of absolute value for the three cases in <xref ref="example-triangle-inequality" text="type-global" />.
</statement>
<answer><p><m>\abs{3}\abs{5}=15=\abs{3\cdot 5} </m>; 
<m>\abs{-3}\abs{-5}=15=\abs{(-3)\cdot (-5)}</m>;
<m>\abs{3}\abs{-5}=15=\abs{3(-5)}</m>
</p></answer>
</exercise>
</subsection>


<subsection><title>Functions Defined Piecewise</title>
    <p>
        A function may be defined by different formulas on different portions of the <m>x</m>-axis. Such a function is said to be defined <term>piecewise</term><idx>piecewise</idx>. To graph a function defined piecewise, we consider each piece of the <m>x</m>-axis separately.
    </p>

<example xml:id="example-piecewise">
    <p>
        Graph the function defined by   
        <me>
            f(x) =
            \begin{cases}
            x +1  \amp \text{if } x\le 1\\
            3  \amp \text{if } x\gt 1
            \end{cases}
        </me>
    </p>
<solution>
    <p>
        Think of the plane as divided into two regions by the vertical line <m>x = 1</m>, as shown in <xref ref="fig-piecewise" text="type-global" />. In the left-hand region (<m>x \le 1</m>), we graph the line <m>y = x + 1</m>. (The fastest way to graph the line is to plot its intercepts, <m>(-1, 0)</m> and <m>(0, 1)</m>.) 
    </p>
    <p></p>

    <sidebyside widths="50% 45%">
    <paragraphs><p>
        Notice that the value <m>x = 1</m> is included in the first region, so <m>f (1) = 1 + 1 = 2</m>, and the point <m>(1, 2)</m> is included on the graph. We indicate this with a solid dot at the point <m>(1, 2)</m>.</p> 

        <p>In the right-hand region (<m>x \gt 1</m>), we graph the horizontal line <m>y = 3</m>. The value <m>x = 1</m> is not included in the second region, so the point <m>(1, 3)</m> is not part of the graph. We indicate this with an open circle at the point <m>(1, 3)</m>.</p>
    </paragraphs>
    <figure xml:id="fig-piecewise"><caption></caption><image source="images/fig-piecewise"><description>piecewise function</description></image></figure>
    </sidebyside>
</solution></example>

<exercise xml:id="exercise-piecewise"><statement>
    Graph the piecewise defined function
    <me>
        g(x) =\begin{cases}
        -1 - x \amp \text { if } x \le -1\\
        x^3 \amp \text{ if } x \gt -1
        \end{cases}
    </me>
</statement>
<answer><p><image source="images/fig-in-ex-ans-2-2-3"  width="30%"><description>piercewise defined function</description></image> </p></answer>
</exercise>

<p>
    The absolute value function <m>f (x) = \abs{x}</m> is an example of a function that is defined piecewise.
    <me>
        f (x) =\abs{x} =
        \begin{cases}
        x \amp \text{if } x\ge 0\\
        -x  \amp \text{if } x\lt 0
        \end{cases}
    </me>
    To sketch the absolute value function, we graph the line <m>y = x</m> in the first quadrant and the line <m>y = -x</m> in the second quadrant.
</p>

<example xml:id="example-piecewise-def">
<ol label="a">
    <li><p>Write a piecewise definition for <m>g(x) = \abs{x - 3}</m>.</p></li>
    <li><p>Sketch a graph of <m>g(x) = \abs{x - 3}</m>.</p></li>
</ol>

<solution>
    <ol label="a">
        <li><p>In the definition for <m>\abs{x}</m>, we replace <m>x</m> by <m>x - 3</m> to get
            <me>
                g(x) =\abs{x-3} =
                \begin{cases}
                x-3 \amp \text{if }~ x-3\ge 0\\
                -x  \amp \text{if }~ -(x-3)\lt 0
                \end{cases}
            </me>
        We can simplify this expression to
            <me>
                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}
            </me>
        </p></li>
        <li><p>
            In the first region, <m>x \ge 3</m>, we graph the line <m>y = x - 3</m>. Because <m>x = 3</m> is included in this region, the endpoint of this portion of the graph, <m>(3, 0)</m>, is included, too.</p>
            <p> In the second region, <m>x \lt 3</m>, we graph the line <m>y = -x + 3</m>. Note that the two pieces of the graph meet at the point <m>(0, 3)</m>, as shown in <xref ref="fig-piecewise2" text="type-global" />.
            <figure xml:id="fig-piecewise2"><caption></caption><image source="images/fig-piecewise2"  width="50%"><description>piecewise function</description></image></figure>
        </p></li>
    </ol>
</solution>
</example>

<exercise xml:id="exercise-piecewise2"><statement>
    <ol label="a">
        <li><p>Use your calculator to graph <m>g(x) =\abs{x-3}</m> and <m>h(x) =\abs{x} + \abs{-3}</m>. Are the graphs the same?</p></li>
        <li><p>Explain why the functions <m>f (x) =\abs{x + k}</m> and <m>g(x) =\abs{x} + \abs{k}</m> are not the same if <m>k\ne 0</m>.
        </p></li>
    </ol>
</statement>
<answer><p><ol label="a">
    <li><p>No</p></li>
    <li><p>Because <m>\abs{x + k}\ne \abs{x} + \abs{k}</m> when <m>x</m> and <m>k</m> have opposite signs</p></li>
</ol></p></answer>
</exercise>

</subsection>
<xi:include href="./summary-2-2.xml" /> <!-- summary  -->
<xi:include href="./section-2-2-exercises.xml" /> <!-- exercises  -->

</section> 
<!-- </book>  </mathbook> -->