Mathematics 265 Introduction to Calculus I

Study Guide :: Unit 2

Functions

The Graph of a Function

To aid our understanding of a function, we resort to its graph whenever possible. The graph of a function is its visual representation. We must learn how to obtain information about a function from its graph; that is, visually.

As we stated above, a function is a set of pairs

$\left\{\left(a,F\left(a\right)\right)|a\text{\hspace{0.17em} is in the domain of \hspace{0.17em}}F\right\}\text{.}$

We can plot the pairs $\left(a,F\left(a\right)\right)$ on the Cartesian plane, with $a$ as the $x$-coordinate and $F\left(a\right)$ as the $y$-coordinate. If we could do this for every number $a$ in the domain of the function $F$, we would end up with a set of points on the Cartesian plane—as many points as the number of elements in the domain. Taken together, all of these points make up the graph of the function $F\text{.}$

You can see that if the domain of $F$ is an infinite set, we cannot possibly plot all the pairs $\left(a,F\left(a\right)\right)$ to obtain the graph of the function $F$. We will learn later how to overcome this problem. For now, we want to learn to read a function’s graph; that is, to obtain information about the function from its graph. Later, we will also learn to graph functions.

If a pair $\left(a,b\right)$ is on the graph of a function $F$, then $a$ is in the domain of $F$ and $F\left(a\right)=b\text{.}$ Observe that the domain of the function $F$ is all numbers $a$ on the $x$-axis, such that $\left(a,b\right)$ is on the graph of $F$ for some $b$ on the $y$-axis.

Example 2.33. See the graph below.

Figure 2.3. Constant function $d$

Figure 2.3 is the graph of a function, say $C$, represented by an infinite horizontal line passing through the point $\left(0,d\right)\text{.}$ [Why?] All of the points on this line are of the form $\left(a,d\right)\text{,}$ with $d$ fixed and $a$ any real number. Since any of these pairs is on the graph of the function $C$, we understand two things: the domain of $C$ is all real numbers, and $C\left(a\right)=d$ for any real number $a$. This function is called the constant function $d$, because the value of the function is $d$ for any real number.

Example 2.34. Compare the graph in Figure 2.3, above, with the following graph:

Figure 2.4. Infinite horizontal line with domain $a<6$

This function, say $P$, also describes an infinite horizontal line, but the line stops at the point $\left(6,C\right)\text{.}$ Any point on this line is of the form $\left(a,C\right)\text{,}$ but $a<6$ (the “open” circle indicates that there is no number on the $y$-axis associated to $6$). That is, the pair $\left(6,b\right)$ is not in the graph of $P$ for any $b$, and we say that $P$ is not defined at $6$. Hence, it is not true that $P\left(6\right)=C\text{.}$ From this graph, we can see that the domain of $P$ is all $a<6$ [in interval notation $\left(-\infty ,6\right)$], and that $P\left(a\right)=C$ for any $a<6$.

Hence,

${\displaystyle P\left(-\frac{3}{4}\right)=C,P\left(3\right)=C,\text{and}P\left(9\right)\text{is undefined}.}$

Example 2.35. Let $Q$ be the function whose graph is shown in Figure 2.5, below.

Figure 2.5. Horizontal line with domain $-6\le a<10$

The graph of $Q$ is a horizontal line from the point $\left(-6,C\right)$ to the point $\left(10,C\right)\text{.}$ The graph of $Q$ has infinitely many points, and the “closed” circle indicates that the pair $\left(-6,C\right)$ is on the graph of $Q$. As you have probably concluded from the open circle, the pair $\left(10,C\right)$ is not on the graph of $Q$. The pairs on this line are of the form $\left(a,C\right)$ with $-6\le a<10$. Hence, the domain of $Q$ is the interval $\left[-6,10\right)\text{.}$ You can see that $Q\left(-6\right)=C\text{,}$ $Q\left(0\right)=C\text{,}$ and $Q\left(10\right)$ is undefined.

Example 2.36. The graph of the function $I$ is as follows:

Figure 2.6. Identity function $I$

This line makes an angle of $45\text{°}$ with the $x$-axis. Any pair on this line has coordinates $\left(a,a\right)\text{.}$ Hence, the domain is all real numbers and $I\left(a\right)=a$ for any real number $a\text{.}$ For example, $I\left(4∕7\right)=4∕7\text{,}$ and $I\left(\pi \right)=\pi \text{.}$

This function is called the identity function.

Example 2.37. Let us study the graph of the function $f$ shown in Figure 2.7, below.

Figure 2.7. Function $f$, Example 2.37

First, let us see which pairs are on the graph and which are not. Are the pairs $\left(0,2\right)$ and $\left(1,2\right)$ on the graph? What is the domain of the function graphed? What is the value of $f\left(1\right)\text{?}$

The function has the constant value $2$ for $x>1$, that is $f\left(x\right)=2$ for $x>1$. The pair $\left(0,1\right)$ is on the graph, but $\left(0,2\right)$ is not; hence, $f\left(0\right)=1\text{.}$ The pair $\left(1,1\right)$ is on the graph, but $\left(1,2\right)$ is not; hence, $f\left(1\right)=1\text{.}$ For any $x$-coordinate there is a $y$-coordinate, such that $\left(x,y\right)$ is on the graph. To see that the domain is all of the real numbers, take any number $x$ and draw a vertical line through the point $\left(x,0\right)\text{,}$ you see that this line cuts the graph of the function at one point, $\left(x,y\right)\text{.}$ This point of intersection is on the graph, and the value of $f$ at $x$ is $y$; that is $f\left(x\right)=y\text{.}$ See Figure 2.8, below.

Figure 2.8. Function $f$, showing intersection with the vertical line through $x$

Example 2.38. Study the following graph of the function $G$, and check that the information given is correct.

Figure 2.9. Function $G$, Example 2.38

The pairs $\left(-4,2\right)\text{,}$ $\left(-4,4\right)$ and $\left(1,4\right)$ are not on the graph, but the pair $\left(1,3\right)$ is on the graph. The domain is all real numbers except $-4.$

$G\left(a\right)=2\text{for}a<-4,G\left(a\right)=4\text{for}-4<a<1,$

$G\left(a\right)=-1\text{for}a>1,\text{and}G\left(1\right)=3.$

Example 2.39. Consider the following graph of a function $f$:

Figure 2.10. Function $f$, Example 2.39

The domain of the function is all positive numbers; that is the interval $\left(0,\infty \right)\text{,}$ the value of the function at $4$ is $0\text{.}$

In a function, a pair $\left(a,b\right)$ is a point on the Cartesian plane. Observe that pairs of the form $\left(a,b\right)$ and $\left(a,c\right)$ are on the vertical line $x=a$, as shown in Figure 2.11, below.

Figure 2.11. Multiple points on the vertical line $x=a$

A function does not include pairs like $\left(a,b\right)$ and $\left(a,c\right)\text{,}$ so the graph of a function does not include such pairs either.

Example 2.40. The following curves are not graphs of functions, because all of them fail the vertical line test.

Figure 2.12. Curves that do not meet the vertical line test

Example 2.41. The following curves are graphs of functions. You can see that no vertical line intersects a curve at more than one point.

Figure 2.13. Curves that meet the vertical line test

To continue our study of functions, we must memorize a few basic graphs. For now, we will accept them as given; later, we will understand why the graphs are the way they are.

Note: In addition to the graphs shown in Table 2.1, below, consider the graphs of the trigonometric functions (shown on page 2 of the “Reference Pages” at the front of the textbook). Study each of these twelve graphs carefully: look for the domain of each graph, and identify some of its points.

Function		Graph
$F(x) = x$ Identity function
$F(x) = {x^2}$ Basic quadratic function${}^{†}$
$F(x) = {x^3}$ Basic cubic function
$F(x) = |x|$ Absolute value function
$F(x) = \dfrac{1}{x}$
$F(x) = \sqrt x $ Basic square root function

${}^{†}$We refer to the function $F(x) = {x^2}$ as the “basic” quadratic function to distinguish it from other quadratic functions, such as $G(x) = {x^2} + 4$. We do the same for the cubic and square root functions.

Table 2.1. Graphs of basic functions