Determining Whether a Relation Represents a Function

A relation is a set of ordered pairs. The set of the first components of each ordered pair is called the domain and the set of the second components of each ordered pair is called the range. Consider the following set of ordered pairs. The first numbers in each pair are the first five natural numbers. The second number in each pair is twice that of the first.

The domain is $\{1,2,3,4,5\}.$ The range is $\{2,4,6,8,10\}.$

Note that each value in the domain is also known as an input value, or independent variable, and is often labeled with the lowercase letter $x.$ Each value in the range is also known as an output value, or dependent variable, and is often labeled lowercase letter $y.$

A function $f$ is a relation that assigns a single value in the range to each value in the domain. In other words, no x-values are repeated. For our example that relates the first five natural numbers to numbers double their values, this relation is a function because each element in the domain, $\{1,2,3,4,5\},$ is paired with exactly one element in the range, $\{2,4,6,8,10\}.$

Now let’s consider the set of ordered pairs that relates the terms “even” and “odd” to the first five natural numbers. It would appear as

Notice that each element in the domain, $\left\{\text{even,}\text{odd}\right\}$ is not paired with exactly one element in the range, $\{1,2,3,4,5\}.$ For example, the term “odd” corresponds to three values from the range, $\{1,3,5\}$ and the term “even” corresponds to two values from the range, $\{2,4\}.$ This violates the definition of a function, so this relation is not a function.

Figure 1 compares relations that are functions and not functions.

Example 1

Determining If Menu Price Lists Are Functions

The coffee shop menu, shown in Figure 2 consists of items and their prices.

1. ⓐ Is price a function of the item?
2. ⓑ Is the item a function of the price?

Solution

1. ⓐ Let’s begin by considering the input as the items on the menu. The output values are then the prices. See Figure 3.

   

   Each item on the menu has only one price, so the price is a function of the item.

2. ⓑ Two items on the menu have the same price. If we consider the prices to be the input values and the items to be the output, then the same input value could have more than one output associated with it. See Figure 4.

   

   Therefore, the item is a not a function of price.

Using Function Notation

Once we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into computers. There are various ways of representing functions. A standard function notation is one representation that facilitates working with functions.

To represent “height is a function of age,” we start by identifying the descriptive variables $h$ for height and $a$ for age. The letters $f,g,$ and $h$ are often used to represent functions just as we use $x,y,$ and $z$ to represent numbers and $A,B,$ and $C$ to represent sets.

$$\begin{array}{lllll}h\text{ is }f\text{ of }a\hfill & \hfill & \hfill & \hfill & \text{We name the function }f;\text{ height is a function of age}.\hfill \\ h=f(a)\hfill & \hfill & \hfill & \hfill & \text{We use parentheses to indicate the function input}\text{. }\hfill \\ f(a)\hfill & \hfill & \hfill & \hfill & \text{We name the function }f;\text{ the expression is read as “}f\text{ of }a\text{.”}\hfill \end{array}$$

Remember, we can use any letter to name the function; the notation $h\left(a\right)$ shows us that $h$ depends on $a.$ The value $a$ must be put into the function $h$ to get a result. The parentheses indicate that age is input into the function; they do not indicate multiplication.

We can also give an algebraic expression as the input to a function. For example $f\left(a+b\right)$ means “first add a and b, and the result is the input for the function f.” The operations must be performed in this order to obtain the correct result.

Example 3

Using Function Notation for Days in a Month

Use function notation to represent a function whose input is the name of a month and output is the number of days in that month. Assume that the domain does not include leap years.

Solution

The number of days in a month is a function of the name of the month, so if we name the function $f,$ we write $\text{days}=f(\text{month})$ or $d=f(m).$ The name of the month is the input to a “rule” that associates a specific number (the output) with each input.

For example, $f\left(\text{March}\right)=31,$ because March has 31 days. The notation $d=f\left(m\right)$ reminds us that the number of days, $d$ (the output), is dependent on the name of the month, $m$ (the input).

Analysis

Note that the inputs to a function do not have to be numbers; function inputs can be names of people, labels of geometric objects, or any other element that determines some kind of output. However, most of the functions we will work with in this book will have numbers as inputs and outputs.

Example 4

Interpreting Function Notation

A function $N=f\left(y\right)$ gives the number of police officers, $N,$ in a town in year $y.$ What does $f\left(2005\right)=300$ represent?

Solution

When we read $f\left(2005\right)=300,$ we see that the input year is 2005. The value for the output, the number of police officers $\left(N\right),$ is 300. Remember, $N=f\left(y\right).$ The statement $f\left(2005\right)=300$ tells us that in the year 2005 there were 300 police officers in the town.

Finding Input and Output Values of a Function

When we know an input value and want to determine the corresponding output value for a function, we evaluate the function. Evaluating will always produce one result because each input value of a function corresponds to exactly one output value.

When we know an output value and want to determine the input values that would produce that output value, we set the output equal to the function’s formula and solve for the input. Solving can produce more than one solution because different input values can produce the same output value.

Evaluation of Functions in Algebraic Forms

When we have a function in formula form, it is usually a simple matter to evaluate the function. For example, the function $f\left(x\right)=5-3x^2$ can be evaluated by squaring the input value, multiplying by 3, and then subtracting the product from 5.

Example 6

Evaluating Functions at Specific Values

Evaluate $f\left(x\right)=x^2+3x-4$ at:

1. ⓐ $2$
2. ⓑ $a$
3. ⓒ $a+h$
4. ⓓ Now evaluate $\frac{f\left(a+h\right)-f\left(a\right)}{h}$

Solution

Replace the $x$ in the function with each specified value.

1. ⓐ Because the input value is a number, 2, we can use simple algebra to simplify.
   $$\begin{array}{l}f\left(2\right)=2^2+3\left(2\right)-4\hfill \\ =4+6-4\hfill \\ =6\hfill \end{array}$$
2. ⓑ In this case, the input value is a letter so we cannot simplify the answer any further.
   $$f\left(a\right)=a^2+3a-4$$
3. ⓒ With an input value of $a+h,$ we must use the distributive property.
   $$\begin{array}{l}f(a+h)={(a+h)}^2+3(a+h)-4\hfill \\ =a^2+2ah+h^2+3a+3h-4\hfill \end{array}$$
4. ⓓ In this case, we apply the input values to the function more than once, and then perform algebraic operations on the result. We already found that
   $$f\left(a+h\right)=a^2+2ah+h^2+3a+3h-4$$

   and we know that

   $$f\left(a\right)=a^2+3a-4$$

   Now we combine the results and simplify.

   $$\begin{array}{l}\begin{array}{l}\hfill \\ \frac{f(a+h)-f(a)}{h}=\frac{(a^2+2ah+h^2+3a+3h-4)-(a^2+3a-4)}{h}\hfill \end{array}\hfill \\ \begin{array}{ll}=\frac{2ah+h^2+3h}{h}\hfill & \hfill \\ =\frac{h(2a+h+3)}{h}\hfill & \begin{array}{cc}\begin{array}{cc}& \end{array}& \end{array}\text{Factor out }h.\hfill \\ =2a+h+3\hfill & \begin{array}{cc}\begin{array}{cc}& \end{array}& \end{array}\text{Simplify}.\hfill \end{array}\hfill \end{array}$$

Example 7

Evaluating Functions

Given the function $h\left(p\right)=p^2+2p,$ evaluate $h\left(4\right).$

Solution

To evaluate $h\left(4\right),$ we substitute the value 4 for the input variable $p$ in the given function.

$$\begin{array}{l}h(p)=p^2+2p\hfill \\ h(4)={(4)}^2+2(4)\hfill \\ =16+8\hfill \\ =24\hfill \end{array}$$

Therefore, for an input of 4, we have an output of 24.

Example 8

Solving Functions

Given the function $h\left(p\right)=p^2+2p,$ solve for $h\left(p\right)=3.$

Solution

$$\begin{array}{llll}h(p)=3\hfill & \hfill & \hfill & \hfill \\ p^2+2p=3\hfill & \hfill & \hfill & \text{Substitute the original function }h(p)=p^2+2p.\hfill \\ p^2+2p-3=0\hfill & \hfill & \hfill & \text{Subtract 3 from each side}.\hfill \\ (p+3\text{)(}p-1)=0\hfill & \hfill & \hfill & \text{Factor}.\hfill \end{array}$$

If $\left(p+3\right)\left(p-1\right)=0,$ either $\left(p+3\right)=0$ or $\left(p-1\right)=0$ (or both of them equal 0). We will set each factor equal to 0 and solve for $p$ in each case.

$$\begin{array}{ll}(p+3)=0,\hfill & p=-3\hfill \\ (p-1)=0,\hfill & p=1\hfill \end{array}$$

This gives us two solutions. The output $h\left(p\right)=3$ when the input is either $p=1$ or $p=-3.$ We can also verify by graphing as in Figure 6. The graph verifies that $h\left(1\right)=h\left(-3\right)=3$ and $h\left(4\right)=24.$

Finding Function Values from a Graph

Evaluating a function using a graph also requires finding the corresponding output value for a given input value, only in this case, we find the output value by looking at the graph. Solving a function equation using a graph requires finding all instances of the given output value on the graph and observing the corresponding input value(s).

Example 12

Reading Function Values from a Graph

Given the graph in Figure 7,

1. ⓐ Evaluate $f\left(2\right).$
2. ⓑ Solve $f\left(x\right)=4.$

Solution

1. ⓐ To evaluate $f\left(2\right),$ locate the point on the curve where $x=2,$ then read the y-coordinate of that point. The point has coordinates $(2,1),$ so $f\left(2\right)=1.$ See Figure 8.

   

2. ⓑ To solve $f\left(x\right)=\mathrm{4,}$ we find the output value $4$ on the vertical axis. Moving horizontally along the line $y=4,$ we locate two points of the curve with output value $\mathrm{4:}$ $(\mathrm{-1},4)$ and $(3,4).$ These points represent the two solutions to $f\left(x\right)=\mathrm{4:}$ $\mathrm{-1}$ or $3.$ This means $f\left(\mathrm{-1}\right)=4$ and $f\left(3\right)=4,$ or when the input is $\mathrm{-1}$ or $\text{3,}$ the output is $\text{4}\text{.}$ See Figure 9.

   

Determining Whether a Function is One-to-One

Some functions have a given output value that corresponds to two or more input values. For example, in the stock chart shown in the figure at the beginning of this chapter, the stock price was $1000 on five different dates, meaning that there were five different input values that all resulted in the same output value of $1000.

However, some functions have only one input value for each output value, as well as having only one output for each input. We call these functions one-to-one functions. As an example, consider a school that uses only letter grades and decimal equivalents, as listed in Table 12.

This grading system represents a one-to-one function, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.

To visualize this concept, let’s look again at the two simple functions sketched in Figure 1(a) and Figure 1(b). The function in part (a) shows a relationship that is not a one-to-one function because inputs $q$ and $r$ both give output $n.$ The function in part (b) shows a relationship that is a one-to-one function because each input is associated with a single output.

Using the Vertical Line Test

As we have seen in some examples above, we can represent a function using a graph. Graphs display a great many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. By convention, graphs are typically constructed with the input values along the horizontal axis and the output values along the vertical axis.

The most common graphs name the input value $x$ and the output value $y,$ and we say $y$ is a function of $x,$ or $y=f\left(x\right)$ when the function is named $f.$ The graph of the function is the set of all points $(x,y)$ in the plane that satisfies the equation $y=f\left(x\right).$ If the function is defined for only a few input values, then the graph of the function is only a few points, where the x-coordinate of each point is an input value and the y-coordinate of each point is the corresponding output value. For example, the black dots on the graph in Figure 10 tell us that $f\left(0\right)=2$ and $f\left(6\right)=1.$ However, the set of all points $(x,y)$ satisfying $y=f\left(x\right)$ is a curve. The curve shown includes $(0,2)$ and $(6,1)$ because the curve passes through those points.

The vertical line test can be used to determine whether a graph represents a function. If we can draw any vertical line that intersects a graph more than once, then the graph does not define a function because a function has only one output value for each input value. See Figure 11.

Example 14

Applying the Vertical Line Test

Which of the graphs in Figure 12 represent(s) a function $y=f\left(x\right)?$

Solution

If any vertical line intersects a graph more than once, the relation represented by the graph is not a function. Notice that any vertical line would pass through only one point of the two graphs shown in parts (a) and (b) of Figure 12. From this we can conclude that these two graphs represent functions. The third graph does not represent a function because, at most x-values, a vertical line would intersect the graph at more than one point, as shown in Figure 13.

Using the Horizontal Line Test

Once we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the horizontal line test. Draw horizontal lines through the graph. If any horizontal line intersects the graph more than once, then the graph does not represent a one-to-one function.

Example 15

Applying the Horizontal Line Test

Consider the functions shown in Figure 12(a) and Figure 12(b). Are either of the functions one-to-one?

Solution

The function in Figure 12(a) is not one-to-one. The horizontal line shown in Figure 15 intersects the graph of the function at two points (and we can even find horizontal lines that intersect it at three points.)

The function in Figure 12(b) is one-to-one. Any horizontal line will intersect a diagonal line at most once.

Identifying Basic Toolkit Functions

In this text, we will be exploring functions—the shapes of their graphs, their unique characteristics, their algebraic formulas, and how to solve problems with them. When learning to read, we start with the alphabet. When learning to do arithmetic, we start with numbers. When working with functions, it is similarly helpful to have a base set of building-block elements. We call these our “toolkit functions,” which form a set of basic named functions for which we know the graph, formula, and special properties. Some of these functions are programmed to individual buttons on many calculators. For these definitions we will use $x$ as the input variable and $y=f\left(x\right)$ as the output variable.

We will see these toolkit functions, combinations of toolkit functions, their graphs, and their transformations frequently throughout this book. It will be very helpful if we can recognize these toolkit functions and their features quickly by name, formula, graph, and basic table properties. The graphs and sample table values are included with each function shown in Table 13.

Table 13 The title is “Toolkit functions”. There are three columns and ten rows. The first column is labeled, “name”, the second column is labeled, “function”, and the third column is labeled graph which contains pictures of the functions. The constant function is f(x) = c where c is the constant; the identity function is f(x) = x; the absolute function is f(x)=|x|; the quadratic function is f(x) = x^2; the cubic function is f(x)=x^3; the reciprocal function is f(x)=1/x; the reciprocal squared function is f(x)=1/x^2; the square root function is f(x)=sqrt(x); the cube root function is f(x) = x^(1/3).

Toolkit Functions
Name	Function	Graph
Constant	$f\left(x\right)=c,$ where $c$ is a constant
Identity	$f\left(x\right)=x$
Absolute value	$f\left(x\right)=\left|x\right|$
Quadratic	$f\left(x\right)=x^2$
Cubic	$f\left(x\right)=x^3$
Reciprocal	$f\left(x\right)=\frac{1}{x}$
Reciprocal squared	$f\left(x\right)=\frac{1}{x^2}$
Square root	$f\left(x\right)=\sqrt{x}$
Cube root	$f\left(x\right)=\sqrt[3]{x}$

Finding the Domain of a Function Defined by an Equation

In Functions and Function Notation, we were introduced to the concepts of domain and range. In this section, we will practice determining domains and ranges for specific functions. Keep in mind that, in determining domains and ranges, we need to consider what is physically possible or meaningful in real-world examples, such as tickets sales and year in the horror movie example above. We also need to consider what is mathematically permitted. For example, we cannot include any input value that leads us to take an even root of a negative number if the domain and range consist of real numbers. Or in a function expressed as a formula, we cannot include any input value in the domain that would lead us to divide by 0.

We can visualize the domain as a “holding area” that contains “raw materials” for a “function machine” and the range as another “holding area” for the machine’s products. See Figure 2.

We can write the domain and range in interval notation, which uses values within brackets to describe a set of numbers. In interval notation, we use a square bracket [ when the set includes the endpoint and a parenthesis ( to indicate that the endpoint is either not included or the interval is unbounded. For example, if a person has $100 to spend, they would need to express the interval that is more than 0 and less than or equal to 100 and write $\left(0,100\right].$ We will discuss interval notation in greater detail later.

Let’s turn our attention to finding the domain of a function whose equation is provided. Oftentimes, finding the domain of such functions involves remembering three different forms. First, if the function has no denominator or an odd root, consider whether the domain could be all real numbers. Second, if there is a denominator in the function’s equation, exclude values in the domain that force the denominator to be zero. Third, if there is an even root, consider excluding values that would make the radicand negative.

Before we begin, let us review the conventions of interval notation:

• The smallest term from the interval is written first.
• The largest term in the interval is written second, following a comma.
• Parentheses, ( or ), are used to signify that an endpoint is not included, called exclusive.
• Brackets, [ or ], are used to indicate that an endpoint is included, called inclusive.

See Figure 3 for a summary of interval notation.

Example 3

Finding the Domain of a Function Involving a Denominator

Find the domain of the function $f(x)=\frac{x+1}{2-x}.$

Solution

When there is a denominator, we want to include only values of the input that do not force the denominator to be zero. So, we will set the denominator equal to 0 and solve for $x.$

$$\begin{array}{l}2-x=0\hfill \\ -x=-2\hfill \\ x=2\hfill \end{array}$$

Now, we will exclude 2 from the domain. The answers are all real numbers where $x<2$ or $x>2.$ We can use a symbol known as the union, $\cup ,$ to combine the two sets. In interval notation, we write the solution: $\left(\mathrm{-\infty },2\right)\cup \left(2,\infty \right).$

In interval form, the domain of $f$ is $\left(-\infty ,2\right)\cup \left(2,\infty \right).$

Using Notations to Specify Domain and Range

In the previous examples, we used inequalities and lists to describe the domain of functions. We can also use inequalities, or other statements that might define sets of values or data, to describe the behavior of the variable in set-builder notation. For example, $\left\{x|10\le x<30\right\}$ describes the behavior of $x$ in set-builder notation. The braces $\left\{\right\}$ are read as “the set of,” and the vertical bar | is read as “such that,” so we would read $\left\{x|10\le x<30\right\}$ as “the set of x-values such that 10 is less than or equal to $x,$ and $x$ is less than 30.”

Figure 5 compares inequality notation, set-builder notation, and interval notation.

To combine two intervals using inequality notation or set-builder notation, we use the word “or.” As we saw in earlier examples, we use the union symbol, $\cup ,$ to combine two unconnected intervals. For example, the union of the sets $\{2,3,5\}$ and $\{4,6\}$ is the set $\{2,3,4,5,6\}.$ It is the set of all elements that belong to one or the other (or both) of the original two sets. For sets with a finite number of elements like these, the elements do not have to be listed in ascending order of numerical value. If the original two sets have some elements in common, those elements should be listed only once in the union set. For sets of real numbers on intervals, another example of a union is

Example 5

Describing Sets on the Real-Number Line

Describe the intervals of values shown in Figure 6 using inequality notation, set-builder notation, and interval notation.

Solution

To describe the values, $x,$ included in the intervals shown, we would say, “ $x$ is a real number greater than or equal to 1 and less than or equal to 3, or a real number greater than 5.”

..

Inequality	$1\le x\le 3\text{or}x>5$
Set-builder notation	$\left\{x|1\le x\le 3\text{or}x>5\right\}$
Interval notation	$[1,3]\cup (5,\infty )$

Remember that, when writing or reading interval notation, using a square bracket means the boundary is included in the set. Using a parenthesis means the boundary is not included in the set.

Finding Domain and Range from Graphs

Another way to identify the domain and range of functions is by using graphs. Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the x-axis. The range is the set of possible output values, which are shown on the y-axis. Keep in mind that if the graph continues beyond the portion of the graph we can see, the domain and range may be greater than the visible values. See Figure 8.

We can observe that the graph extends horizontally from $\mathrm{-5}$ to the right without bound, so the domain is $\left[\mathrm{-5},\infty \right).$ The vertical extent of the graph is all range values $5$ and below, so the range is $\left(\mathrm{-\infty },5\right].$ Note that the domain and range are always written from smaller to larger values, or from left to right for domain, and from the bottom of the graph to the top of the graph for range.

Example 6

Finding Domain and Range from a Graph

Find the domain and range of the function $f$ whose graph is shown in Figure 9.

Solution

We can observe that the horizontal extent of the graph is –3 to 1, so the domain of $f$ is $\left(-3,1\right].$

The vertical extent of the graph is 0 to –4, so the range is $[-4,0].$ See Figure 10.

Example 7

Finding Domain and Range from a Graph of Oil Production

Find the domain and range of the function $f$ whose graph is shown in Figure 11.

Solution

The input quantity along the horizontal axis is “years,” which we represent with the variable $t$ for time. The output quantity is “thousands of barrels of oil per day,” which we represent with the variable $b$ for barrels. The graph may continue to the left and right beyond what is viewed, but based on the portion of the graph that is visible, we can determine the domain as $1973\le t\le 2008$ and the range as approximately $180\le b\le 2010.$

In interval notation, the domain is [1973, 2008], and the range is about [180, 2010]. For the domain and the range, we approximate the smallest and largest values since they do not fall exactly on the grid lines.

Finding Domains and Ranges of the Toolkit Functions

We will now return to our set of toolkit functions to determine the domain and range of each.

Example 10

Finding the Domain and Range

Find the domain and range of $f(x)=2\sqrt{x+4}.$

Solution

We cannot take the square root of a negative number, so the value inside the radical must be nonnegative.

$x+4\ge 0\text{ when }x\ge -4$

The domain of $f\left(x\right)$ is $[-4,\infty ).$

We then find the range. We know that $f\left(-4\right)=0,$ and the function value increases as $x$ increases without any upper limit. We conclude that the range of $f$ is $[0,\infty ).$

Analysis

Figure 22 represents the function $f.$

Graphing Piecewise-Defined Functions

Sometimes, we come across a function that requires more than one formula in order to obtain the given output. For example, in the toolkit functions, we introduced the absolute value function $f(x)=\left|x\right|.$ With a domain of all real numbers and a range of values greater than or equal to 0, absolute value can be defined as the magnitude, or modulus, of a real number value regardless of sign. It is the distance from 0 on the number line. All of these definitions require the output to be greater than or equal to 0.

If we input 0, or a positive value, the output is the same as the input.

If we input a negative value, the output is the opposite of the input.

Because this requires two different processes or pieces, the absolute value function is an example of a piecewise function. A piecewise function is a function in which more than one formula is used to define the output over different pieces of the domain.

We use piecewise functions to describe situations in which a rule or relationship changes as the input value crosses certain “boundaries.” For example, we often encounter situations in business for which the cost per piece of a certain item is discounted once the number ordered exceeds a certain value. Tax brackets are another real-world example of piecewise functions. For example, consider a simple tax system in which incomes up to $10,000 are taxed at 10%, and any additional income is taxed at 20%. The tax on a total income $S$ would be $0.1S$ if $S\le \text{\$}10\text{,}000$ and $\text{\$}1000+0.2(S-\text{\$}10\text{,}000)$ if $S>\text{\$}10\text{,}000.$

Example 11

Writing a Piecewise Function

A museum charges $5 per person for a guided tour with a group of 1 to 9 people or a fixed $50 fee for a group of 10 or more people. Write a function relating the number of people, $n,$ to the cost, $C.$ Since one cannot have fractions of a person, this is really a discrete function. However, for this exercise we will treat it as a continuous function.

Solution

Two different formulas will be needed. For n-values under 10, $C=5n.$ For values of $n$ that are 10 or greater, $C=50.$

$$C(n)=\{\begin{array}{ccc}5n& \text{if}& 0<n<10\\ 50& \text{if}& n\ge 10\end{array}$$

Analysis

The function is represented in Figure 23. The graph is a diagonal line from $n=0$ to $n=10$ and a constant after that. In this example, the two formulas agree at the meeting point where $n=10,$ but not all piecewise functions have this property.

Example 12

Working with a Piecewise Function

A cell phone company uses the function below to determine the cost, $C,$ in dollars for $g$ gigabytes of data transfer.

$$C(g)=\{\begin{array}{ccc}25& \text{if}& 0<g<2\\ 25+10(g-2)& \text{if}& g\ge 2\end{array}$$

Find the cost of using 1.5 gigabytes of data and the cost of using 4 gigabytes of data.

Solution

To find the cost of using 1.5 gigabytes of data, $C(1.5),$ we first look to see which part of the domain our input falls in. Because 1.5 is less than 2, we use the first formula.

$$C(1.5)=\text{\$}25$$

To find the cost of using 4 gigabytes of data, $C(4),$ we see that our input of 4 is greater than 2, so we use the second formula.

$$C(4)=25+10(4-2)=\text{\$}45$$

Analysis

The function is represented in Figure 24. We can see where the function changes from a constant to a shifted and stretched identity at $g=2.$ We plot the graphs for the different formulas on a common set of axes, making sure each formula is applied on its proper domain.

Example 13

Graphing a Piecewise Function

Sketch a graph of the function.

$$f(x)=\{\begin{array}{ccc}{x}^2& \text{if}& x\le 1\\ 3& \text{if}& 1<x\le 2\\ x& \text{if}& x>2\end{array}$$

Solution

Each of the component functions is from our library of toolkit functions, so we know their shapes. We can imagine graphing each function and then limiting the graph to the indicated domain. At the endpoints of the domain, we draw open circles to indicate where the endpoint is not included because of a less-than or greater-than inequality; we draw a closed circle where the endpoint is included because of a less-than-or-equal-to or greater-than-or-equal-to inequality.

Figure 25 shows the three components of the piecewise function graphed on separate coordinate systems.

Now that we have sketched each piece individually, we combine them in the same coordinate plane. See Figure 26.

Analysis

Note that the graph does pass the vertical line test even at $x=1$ and $x=2$ because the points $\left(1,3\right)$ and $\left(2,2\right)$ are not part of the graph of the function, though $\left(1,1\right)$ and $\left(2,3\right)$ are.