Properties of Functions and Basic Function Types
Site: | Saylor Academy |
Course: | MA001: College Algebra |
Book: | Properties of Functions and Basic Function Types |
Printed by: | Guest user |
Date: | Saturday, September 7, 2024, 8:08 PM |
Description
In this section, you will analyze graphs to determine whether they represent a function and 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 course.
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 and the output value
, and we say
is a function of
, or
when the function is named
. The graph of the function is the set of all points
in the plane that satisfies the equation
. If the function is defined for only a few input values, then the graph of the function is only a few points, where the
-coordinate of each point is an input value and the
-coordinate of each point is the corresponding output value. For example, the black dots on the graph in Figure 7 tell us that
and
. However, the set of all points
satisfying
is a curve. The curve shown includes
and
because the curve passes through those points.
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.
Figure 8
HOW TO
Given a graph, use the vertical line test to determine if the graph represents a function.
1. Inspect the graph to see if any vertical line drawn would intersect the curve more than once.
2. 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
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 -values, a vertical line would intersect the graph at more than one point, as shown in Figure 10.
Figure 10
TRY IT #11
Does the graph in Figure 11 represent a 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.
1. Inspect the graph to see if any horizontal line drawn would intersect the curve more than once.
2. 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).
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 as the input variable and
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 | ![]() |
||
Identity | ![]() |
||
Absolute value | ![]() |
||
Quadratic | ![]() |
||
Cubic | ![]() |
||
Reciprocal | ![]() |
||
Reciprocal squared | ![]() |
||
Square root | ![]() |
||
Cube root | ![]() |
Table 13
Toolkit Functions
Source: Linda Green, https://youtu.be/4L7KxV-G4Fk
This work is licensed under a Creative Commons Attribution 4.0 License.