Identify Functions Using Graphs

Learning Objectives

Verify a function using the vertical line test
Verify a one-to-one function with the horizontal line test
Identify the graphs of the toolkit functions

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 [latex]x[/latex] and the output value [latex]y[/latex], and we say [latex]y[/latex] is a function of [latex]x[/latex], or [latex]y=f\left(x\right)[/latex] when the function is named [latex]f[/latex]. The graph of the function is the set of all points [latex]\left(x,y\right)[/latex] in the plane that satisfies the equation [latex]y=f\left(x\right)[/latex]. 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 the graph below tell us that [latex]f\left(0\right)=2[/latex] and [latex]f\left(6\right)=1[/latex]. However, the set of all points [latex]\left(x,y\right)[/latex] satisfying [latex]y=f\left(x\right)[/latex] is a curve. The curve shown includes [latex]\left(0,2\right)[/latex] and [latex]\left(6,1\right)[/latex] because the curve passes through those points. Graph of a polynomial.

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

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

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: Applying the Vertical Line Test

Which of the graphs represent(s) a function [latex]y=f\left(x\right)?[/latex] Graph of a polynomial.

Answer: 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 the graph above. 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. Graph of a circle.

https://youtu.be/5Z8DaZPJLKY

Try It

Does the graph below represent a function? Graph of absolute value function.

Answer: yes

https://youtu.be/tbSGdcSN8RE

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 [latex]x[/latex] as the input variable and [latex]y=f\left(x\right)[/latex] 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 below.

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

Try It now

In this exercise, you will graph the toolkit functions using Desmos.

Graph each toolkit function using function notation.
Make a table of values that references the function and includes at least the interval [-5,5].

Licenses & Attributions

CC licensed content, Original

Question ID 111715, 11722. Provided by: Lumen Learning License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.
Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
Vertical Line Test. Authored by: Lumen Learning. Located at: https://www.desmos.com/calculator/dcq8twow2q. License: CC BY: Attribution.

CC licensed content, Shared previously

College Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. Located at: https://openstax.org/books/college-algebra/pages/1-introduction-to-prerequisites. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].
Question ID 40676. Authored by: Jenck, Michael. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.

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