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Graphs of Functions

Use the vertical line test

Use the Vertical Line Test

In the last section we learned how to determine if a relation is a function. The relations we looked at were expressed as a set of ordered pairs, a mapping or an equation. We will now look at how to tell if a graph is that of a function.

An ordered pair \((x,y)\) is a solution of a linear equation, if the equation is a true statement when the x- and y-values of the ordered pair are substituted into the equation.

The graph of a linear equation is a straight line where every point on the line is a solution of the equation and every solution of this equation is a point on this line.

In , we can see that, in graph of the equation \(y=2x-3,\) for every x-value there is only one y-value, as shown in the accompanying table.

A relation is a function if every element of the domain has exactly one value in the range. So the relation defined by the equation \(y=2x-3\) is a function.

If we look at the graph, each vertical dashed line only intersects the line at one point. This makes sense as in a function, for every x-value there is only one y-value.

If the vertical line hit the graph twice, the x-value would be mapped to two y-values, and so the graph would not represent a function.

Example

Try it.

Determine whether each graph is the graph of a function.

Solution

ⓐ Since any vertical line intersects the graph in at most one point, the graph is the graph of a function.

ⓑ One of the vertical lines shown on the graph, intersects it in two points. This graph does not represent a function.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Identify Graphs of Basic Functions

We used the equation \(y=2x-3\) and its graph as we developed the vertical line test. We said that the relation defined by the equation \(y=2x-3\) is a function.

We can write this as in function notation as \(f(x)=2x-3.\) It still means the same thing. The graph of the function is the graph of all ordered pairs \((x,y)\) where \(y=f(x).\) So we can write the ordered pairs as \((x,f(x)).\) It looks different but the graph will be the same.

Compare the graph of \(y=2x-3\) previously shown in with the graph of \(f(x)=2x-3\) shown in . Nothing has changed but the notation.

As we move forward in our study, it is helpful to be familiar with the graphs of several basic functions and be able to identify them.

Through our earlier work, we are familiar with the graphs of linear equations. The process we used to decide if \(y=2x-3\) is a function would apply to all linear equations. All non-vertical linear equations are functions. Vertical lines are not functions as the x-value has infinitely many y-values.

We wrote linear equations in several forms, but it will be most helpful for us here to use the slope-intercept form of the linear equation. The slope-intercept form of a linear equation is \(y=mx+b.\) In function notation, this linear function becomes \(f(x)=mx+b\) where m is the slope of the line and b is the y-intercept.

The domain is the set of all real numbers, and the range is also the set of all real numbers.

Example

Try it.

Graph: \(f(x)=-2x-4.\)

Solution

	\(\ f(x)=-2x-4\)
We recognize this as a linear function.
Find the slope and y-intercept.	\(\ m=-2\) \(\ b=-4\)
Graph using the slope intercept.

Example

Try it.

Graph: \(f(x)=4.\)

Solution

	\(f(x)=4\)
We recognize this as a constant function.
The graph will be a horizontal line through \((0,4).\)

Example

Try it.

Graph: \(f(x)={x}^{2}.\)

Solution

We choose x-values. We substitute them in and then create a chart as shown.

Example

Try it.

Graph: \(f(x)={x}^{3}.\)

Solution

We choose x-values. We substitute them in and then create a chart.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Read Information from a Graph of a Function

In the sciences and business, data is often collected and then graphed. The graph is analyzed, information is obtained from the graph and then often predictions are made from the data.

We will start by reading the domain and range of a function from its graph.

Remember the domain is the set of all the x-values in the ordered pairs in the function. To find the domain we look at the graph and find all the values of x that have a corresponding value on the graph. Follow the value x up or down vertically. If you hit the graph of the function then x is in the domain.

Remember the range is the set of all the y-values in the ordered pairs in the function. To find the range we look at the graph and find all the values of y that have a corresponding value on the graph. Follow the value y left or right horizontally. If you hit the graph of the function then y is in the range.

Example

Try it.

Use the graph of the function to find its domain and range. Write the domain and range in interval notation.

Solution

To find the domain we look at the graph and find all the values of x that correspond to a point on the graph. The domain is highlighted in red on the graph. The domain is \([-3,3].\)

To find the range we look at the graph and find all the values of y that correspond to a point on the graph. The range is highlighted in blue on the graph. The range is \([-1,3].\)

We are now going to read information from the graph that you may see in future math classes.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Key Concepts

Vertical Line Test
- A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point.
- If any vertical line intersects the graph in more than one point, the graph does not represent a function.
Graph of a Function
- The graph of a function is the graph of all its ordered pairs, \((x,y)\) or using function notation, \((x,f(x))\) where \(y=f(x).\)
  \[\begin{array}{llll}f & & & \text{name of function} \\ x & & & x\text{-coordinate of the ordered pair} \\ f(x) & & & y\text{-coordinate of the ordered pair}\end{array}\]
Linear Function
Constant Function
Identity Function
Square Function
Cube Function
Square Root Function
Absolute Value Function

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Evaluate: ⓐ \({2}^{3}\) ⓑ \({3}^{2}.\)
If you missed this problem, review .

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ⓐ \(8\); ⓑ \(9\)
Evaluate: ⓐ \(|7|\) ⓑ \(|-3|.\)
If you missed this problem, review .

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ⓐ \(7\); ⓑ \(3\)
Evaluate: ⓐ \(\sqrt{4}\) ⓑ \(\sqrt{16}.\)
If you missed this problem, review .

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ⓐ \(2\); ⓑ \(4\)
Determine whether each graph is the graph of a function.

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ⓐ Since any vertical line intersects the graph in at most one point, the graph is the graph of a function.

ⓑ One of the vertical lines shown on the graph, intersects it in two points. This graph does not represent a function.
Determine whether each graph is the graph of a function.

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ⓐ yes ⓑ no
Determine whether each graph is the graph of a function.

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ⓐ no ⓑ yes
Graph: \(f(x)=-2x-4.\)

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\(\ f(x)=-2x-4\)
We recognize this as a linear function.
Find the slope and y-intercept. \(\ m=-2\)
\(\ b=-4\)
Graph using the slope intercept.
Graph: \(f(x)=-3x-1\)

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Graph: \(f(x)=-4x-5\)

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Graph: \(f(x)=4.\)

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\(f(x)=4\)
We recognize this as a constant function.
The graph will be a horizontal line through \((0,4).\)
Graph: \(f(x)=-2.\)

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Graph: \(f(x)=3.\)

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Graph: \(f(x)={x}^{2}.\)

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We choose x-values. We substitute them in and then create a chart as shown.
Graph: \(f(x)={x}^{2}.\)

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\(f(x)=\text{-}{x}^{2}\)

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Graph: \(f(x)={x}^{3}.\)

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We choose x-values. We substitute them in and then create a chart.
Graph: \(f(x)={x}^{3}.\)

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Graph: \(f(x)=\text{-}{x}^{3}.\)

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\(f(x)=\sqrt{x}\)

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We choose x-values. Since we will be taking the square root, we choose numbers that are perfect squares, to make our work easier. We substitute them in and then create a chart.
Graph: \(f(x)=\sqrt{x}.\)

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Graph: \(f(x)=\text{-}\sqrt{x}.\)

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Graph: \(f(x)=|x|.\)

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We choose x-values. We substitute them in and then create a chart.
Graph: \(f(x)=|x|.\)

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Graph: \(f(x)=\text{-}|x|.\)

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Use the graph of the function to find its domain and range. Write the domain and range in interval notation.

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To find the domain we look at the graph and find all the values of x that correspond to a point on the graph. The domain is highlighted in red on the graph. The domain is \([-3,3].\)
To find the range we look at the graph and find all the values of y that correspond to a point on the graph. The range is highlighted in blue on the graph. The range is \([-1,3].\)
Use the graph of the function to find its domain and range. Write the domain and range in interval notation.

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The domain is \([-5,1].\) The range is \([-4,2].\)
Use the graph of the function to find its domain and range. Write the domain and range in interval notation.

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The domain is \([-2,4].\) The range is \([-5,3].\)
Use the graph of the function to find the indicated values.
ⓐ Find: \(f(0).\)
ⓑ Find: \(f(\frac{3}{2}\pi ).\)
ⓒ Find: \(f(-\frac{1}{2}\pi ).\)
ⓓ Find the values for x when \(f(x)=0.\)
ⓔ Find the x-intercepts.
ⓕ Find the y-intercepts.
ⓖ Find the domain. Write it in interval notation.
ⓗ Find the range. Write it in interval notation.

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ⓐ When \(x=0,\) the function crosses the y-axis at 0. So, \(f(0)=0.\)
ⓑ When \(x=\frac{3}{2}\pi ,\) the y-value of the function is \(-1.\) So, \(f(\frac{3}{2}\pi )=-1.\)
ⓒ When \(x=-\frac{1}{2}\pi ,\) the y-value of the function is \(-1.\) So, \(f(-\frac{1}{2}\pi )=-1.\)
ⓓ The function is 0 at the points, \((-2\pi ,0),(\text{-}\pi ,0),(0,0),(\pi ,0),(2\pi ,0).\) The x-values when \(f(x)=0\) are \(-2\pi ,\text{-}\pi ,0,\pi ,2\pi .\)
ⓔ The x-intercepts occur when \(y=0.\) So the x-intercepts occur when \(f(x)=0.\) The x-intercepts are \((-2\pi ,0),(\text{-}\pi ,0),(0,0),(\pi ,0),(2\pi ,0).\)
ⓕ The y-intercepts occur when \(x=0.\) So the y-intercepts occur at \(f(0).\) The y-intercept is \((0,0).\)
ⓖ This function has a value for all values of x. Therefore, the domain in interval notation is \((-\infty ,\ \infty )\)
ⓗ This function values, or y-values go from \(-1\) to 1. Therefore, the range, in interval notation, is \([-1,1].\)
Use the graph of the function to find the indicated values.
ⓐ Find: \(f(0).\)
ⓑ Find: \(f(\frac{1}{2}\pi ).\)
ⓒ Find: \(f(-\frac{3}{2}\pi ).\)
ⓓ Find the values for x in [\(-2\pi ,2\pi\)] when \(f(x)=0.\)
ⓔ Find the x-intercepts.
ⓕ Find the y-intercepts.
ⓖ Find the domain. Write it in interval notation.
ⓗ Find the range. Write it in interval notation.

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ⓐ \(f(0)=0\) ⓑ \(f=(\frac{\pi }{2})=2\) ⓒ \(f=(\frac{-3\pi }{2})=2\) ⓓ \(f(x)=0\) for \(x=-2\pi ,\text{-}\pi ,0,\pi ,2\pi\) ⓔ \((-2\pi ,0),(\text{-}\pi ,0),(0,0),(\pi ,0),(2\pi ,0)\) ⓕ \((0,0)\) ⓖ \((-\infty ,\ \infty )\) ⓗ \([-2,2]\)
Use the graph of the function to find the indicated values.
ⓐ Find: \(f(0).\)
ⓑ Find: \(f(\pi ).\)
ⓒ Find: \(f(\text{-}\pi ).\)
ⓓ Find the values for x in [\(-2\pi ,2\pi\)] when \(f(x)=0.\)
ⓔ Find the x-intercepts.
ⓕ Find the y-intercepts.
ⓖ Find the domain. Write it in interval notation.
ⓗ Find the range. Write it in interval notation.

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ⓐ \(f(0)=1\) ⓑ \(f(\pi )=-1\) ⓒ \(f(\text{-}\pi )=-1\) ⓓ \(f(x)=0\) for \(x=-\frac{3\pi }{2},-\frac{\pi }{2},\frac{\pi }{2},\frac{3\pi }{2}\) ⓔ \((-\frac{3\pi }{2},0),(-\frac{\pi }{2},0),(\frac{\pi }{2},0),(\frac{3\pi }{2},0)\) ⓕ \((0,1)\) ⓖ \((-\infty ,\ \infty )\) ⓗ \([-1,1]\)
\(f(x)=3x+4\)

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ⓐ

ⓑ D:(-∞,∞), R:(-∞,∞)
\(f(x)=2x+5\)
\(f(x)=\text{-}x-2\)

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ⓐ

ⓑ D:(-∞,∞), R:(-∞,∞)
\(f(x)=-4x-3\)
\(f(x)=-2x+2\)

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ⓐ

ⓑ D:(-∞,∞), R:(-∞,∞)
\(f(x)=-3x+3\)
\(f(x)=\frac{1}{2}x+1\)

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ⓐ

ⓑ D:(-∞,∞), R:(-∞,∞)
\(f(x)=\frac{2}{3}x-2\)
\(f(x)=-2x\)

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ⓐ

ⓑ D:(-∞,∞), R:(-∞,∞)
\(f(x)=-3x\)

Symbols used here

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

\infty

infinity
Not a number: "grows without bound" in limits and intervals.

\pi

pi
Ratio of a circle's circumference to its diameter, 3.14159…

\sup,\ \inf

supremum, infimum
Least upper bound, greatest lower bound.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

imaginary unit
i² = −1.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Graphs of Functions

Use the vertical line test
Identify graphs of basic functions
Read information from a graph of a function
A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point.
If any vertical line intersects the graph in more than one point, the graph does not represent a function.
The graph of a function is the graph of all its ordered pairs,

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

Санҷиши худ

Parts of this page are adapted from OpenStax Intermediate Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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