Properties of Functions and Basic Function Types

Site:	Saylor Academy
Course:	MA120: Applied College Algebra
Book:	Properties of Functions and Basic Function Types

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Date:	Saturday, 3 May 2025, 2:25 PM

Description

This is a book resource with multiple pages. Navigate between the pages using the buttons.

Using the Vertical Line Test
Using the Horizontal Line Test
Identifying Basic Toolkit Functions
Toolkit Functions

Using the Vertical Line Test

In this section, you will analyze graphs to determine whether they represent a function and will be introduced to the graphs of the basic functions. Pay close attention to the basic functions because they will be referred to throughout most of the rest of the course.

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 $y=f(x)$ 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 $y=f(x)$ . 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 7 tell us that $f(0)=2$ and $f(6)=1$ . However, the set of all points $(x, y)$ satisfying $y=f(x)$ is a curve. The curve shown includes $(0,2)$ and $(6,1)$ because the curve passes through those points.

Graph of a polynomial.

Figure 7

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 8.

Three graphs visually showing what is and is not a function.

Figure 8

HOW TO

Given a graph, use the vertical line test to determine if the graph represents a function.

Inspect the graph to see if any vertical line drawn would intersect the curve more than once.
If there is any such line, determine that the graph does not represent a function.

Example 14

Applying the Vertical Line Test

Which of the graphs in Figure 9 represent(s) a function $y=f(x) ?$

Graph of a polynomial.

Figure 9

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 Eigure 9. 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 10.

Graph of a circle.

Figure 10

Try It #11

Does the graph in Figure 11 represent a function?

Graph of absolute value function.

Figure 11

Source: Rice University, https://openstax.org/books/college-algebra/pages/3-1-functions-and-function-notation
This work is licensed under a Creative Commons Attribution 4.0 License.

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.

HOW TO

Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function.

Inspect the graph to see if any horizontal line drawn would intersect the curve more than once.
If there is any such line, determine that the function is not one-to-one.

Example 15

Applying the Horizontal Line Test

Consider the functions shown in Figure 9(a) and Figure 9(b). Are either of the functions one-to-one?

Solution

The function in Figure 9(a) is not one-to-one. The horizontal line shown in Figure 12 intersects the graph of the function at two points (and we can even find horizontal lines that intersect it at three points).

one-to-one. The horizontal line shown in Figure 12 intersects the graph of the function at two points (and we can even find h

Figure 12

The function in Eigure 9(b) is one-to-one. Any horizontal line will intersect a diagonal line at most once.

Try It #12

Is the graph shown in Figure 9 one-to-one?

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 $y=f(x)$ 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.

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