First, please create an account

Already have a Sophia account?

Graphs of Rational Functions

Author: Sophia

what's covered

In this lesson, you will use the characteristics of rational functions in order to graph a rational function. Specifically, this lesson will cover:

1. Intercepts of a Rational Function
2. The Graph of a Rational Function
3. The Range of a Rational Function

1. Intercepts of a Rational Function

Given a function f open parentheses x close parentheses comma recall the following:

The y-intercept of the graph of a function is
The x-intercept(s) of the graph of a function are the points , where x is a solution to

Since rational functions could be undefined at specific values of x, it is possible that the graph of a rational function has no x-intercept or no y-intercept.

hint

Always find the domain of a rational function first. Then compare each characteristic you find with the domain to make sure that all information you find is valid.

EXAMPLE

Consider the function $f open parentheses x close parentheses equals fraction numerator 2 x minus 1 over denominator x minus 4 end fraction.$

Note: this function is undefined when x equals 4.

This means the domain is open parentheses short dash infinity comma space 4 close parentheses union open parentheses 4 comma space infinity close parentheses.

open parentheses short dash infinity comma space 4 close parentheses union open parentheses 4 comma space infinity close parentheses.

Since $f open parentheses 0 close parentheses equals fraction numerator 2 open parentheses 0 close parentheses minus 1 over denominator 0 minus 4 end fraction equals 1 fourth comma$ the y-intercept is open parentheses 0 comma space 1 fourth close parentheses.

open parentheses 0 comma space 1 fourth close parentheses.

To find any x-intercepts, set f open parentheses x close parentheses equals 0

and solve.

$fraction numerator 2 x minus 1 over denominator x minus 4 end fraction equals 0$	Set
$fraction numerator 2 x minus 1 over denominator x minus 4 end fraction open parentheses x minus 4 close parentheses equals 0 open parentheses x minus 4 close parentheses$	Multiply both sides by since
	Simplify.
	Solve for x.

Since

is in the domain of f, the x-intercept is open parentheses 1 half comma space 0 close parentheses.

The graph of f open parentheses x close parentheses

has y-intercept open parentheses 0 comma space 1 fourth close parentheses

and x-intercept open parentheses 1 half comma space 0 close parentheses.

Note that solving $fraction numerator 2 x minus 1 over denominator x minus 4 end fraction equals 0$ requires only that 2 x minus 1 equals 0. This is because $fraction numerator 2 x minus 1 over denominator x minus 4 end fraction$ is in reduced form.

big idea

When solving $fraction numerator N open parentheses x close parentheses over denominator D open parentheses x close parentheses end fraction equals 0 comma$ it is only necessary to solve

N open parentheses x close parentheses equals 0.

Be sure to check that D open parentheses x close parentheses not equal to 0.

This is intuitive since the only way for a fraction to be equal to zero is if its numerator is equal to 0 and its denominator is not equal to 0.

try it

Consider the function $f open parentheses x close parentheses equals fraction numerator 3 x minus 12 over denominator x squared plus x minus 12 end fraction.$

Find the x- and y-intercepts of the graph of f (x ).

First, write f open parentheses x close parentheses

in factored form: $f open parentheses x close parentheses equals fraction numerator 3 open parentheses x minus 4 close parentheses over denominator open parentheses x plus 4 close parentheses open parentheses x minus 3 close parentheses end fraction$

Note that the domain is the set of all real numbers except -4 and 3.

Using interval notation, this is expressed as open parentheses short dash infinity comma space short dash 4 close parentheses union open parentheses short dash 4 comma space 3 close parentheses union open parentheses 3 comma space infinity close parentheses.

open parentheses short dash infinity comma space short dash 4 close parentheses union open parentheses short dash 4 comma space 3 close parentheses union open parentheses 3 comma space infinity close parentheses.

To find any x-intercepts, set the numerator equal to 0 and solve for x:

	Set numerator equal to 0.
	Add 12 to both sides.
	Divide both sides by 3.

Note that 4 is in the domain of f, which means that there is an x-intercept at open parentheses 4 comma space 0 close parentheses.

To find the y-intercept, note again that 0 is in the domain of f, so a y-intercept exists.

$f open parentheses 0 close parentheses equals fraction numerator 3 open parentheses 0 close parentheses minus 12 over denominator 0 squared plus 0 minus 12 end fraction equals fraction numerator short dash 12 over denominator short dash 12 end fraction equals 1$

The y-intercept is open parentheses 0 comma space 1 close parentheses.

2. The Graph of a Rational Function

To graph a rational function, there are several characteristics to be considered:

Vertical asymptotes and holes in the graph
Horizontal, slant, and nonlinear asymptotes
x- and y-intercepts

In order to get a complete graph of the function, more information might be needed. This varies by example.

EXAMPLE

Consider the function $f open parentheses x close parentheses equals fraction numerator 3 x minus 12 over denominator 2 x plus 4 end fraction.$

Note that this function is in simplest form—there is no common factor to remove. The denominator is equal to zero when 2 x plus 4 equals 0 comma

This means the domain of f open parentheses x close parentheses

open parentheses short dash infinity comma space short dash 2 close parentheses union open parentheses short dash 2 comma space infinity close parentheses.

Here are some characteristics of the graph of f open parentheses x close parentheses.

Characteristic	Rationale
The equation of the vertical asymptote is	Since is in simplest form, any real value of x where the denominator is equal to zero corresponds to a vertical asymptote. Setting we get
The equation of the horizontal asymptote is This also means that as and	Since the degrees of the numerator and denominator are equal, has a horizontal asymptote with equation where b is determined by dividing the leading coefficients. This means the equation of the horizontal asymptote is
The x-intercept is	To find the x-intercept, set and solve. This is equivalent to setting the numerator equal to zero. Solving we get Since this value doesn’t also make the denominator equal to 0, the x-intercept is
The y-intercept is	Since $f open parentheses 0 close parentheses equals fraction numerator 3 open parentheses 0 close parentheses minus 12 over denominator 2 open parentheses 0 close parentheses plus 4 end fraction equals short dash 3 comma$ the y-intercept is

Putting this all together, here is the graph of f open parentheses x close parentheses.

A graph with an x-axis ranging from −11 to 12 and a y-axis ranging from −10 to 10. The graph contains a horizontal dashed line with equation y equals 1.5, representing a horizontal asymptote, and a vertical dashed line with equation x equals -2, representing a vertical asymptote. There are two curves, separated by the vertical asymptote. The left curve starts to the extreme left of the second quadrant, slightly above the horizontal asymptote, then as x increases, curves upward, running along the vertical asymptote on the left. The curve on the right starts at the bottom of the lower left quadrant, slightly to the right of the vertical asymptote. As x increases, the curve increases, passing through the points (0, -3) and (4, 0), then increases toward the horizontal asymptote without ever crossing it.

Notice the intercepts and asymptotes in the graph. The other points were obtained by substituting additional values of x into the function and evaluating. It is helpful to plot points on each side of the vertical asymptote. Since both intercepts are on the right of the vertical asymptote, values of x that are less than -2 were chosen to get an idea of the shape of the graph to the left of x equals short dash 2.

Notice how the horizontal asymptote guides the direction of the graph to the extreme left and right, and the vertical asymptote guides the direction of the graph as x gets closer to -2.

Let’s look at another example, this one with two vertical asymptotes.

EXAMPLE

Consider the function $f open parentheses x close parentheses equals fraction numerator 9 x minus 36 over denominator 4 x squared minus 36 end fraction.$

In factored form, $f open parentheses x close parentheses equals fraction numerator 9 open parentheses x minus 4 close parentheses over denominator 4 open parentheses x plus 3 close parentheses open parentheses x minus 3 close parentheses end fraction.$ Since the numerator and denominator share no common factors, there are no holes in the graph of

The domain of

is all real numbers except x equals 3

and

or in interval notation, open parentheses short dash infinity comma space short dash 3 close parentheses union open parentheses short dash 3 comma space 3 close parentheses union open parentheses 3 comma space infinity close parentheses.

open parentheses short dash infinity comma space short dash 3 close parentheses union open parentheses short dash 3 comma space 3 close parentheses union open parentheses 3 comma space infinity close parentheses.

Here are some characteristics of the graph of

Characteristic	Rationale
The equations of the vertical asymptotes are and	Since is in simplest form, any real value of x where the denominator is equal to zero corresponds to a vertical asymptote.
The equation of the horizontal asymptote is This also means that as and	Since the degree of the numerator is less than the degree of the denominator, has horizontal asymptote
The x-intercept is	To find the x-intercept, set and solve. This is equivalent to setting the numerator equal to zero. Solving we get Since this value doesn’t also make the denominator equal to 0, the x-intercept is
The y-intercept is	Since $f open parentheses x close parentheses equals fraction numerator 9 open parentheses 0 close parentheses minus 36 over denominator 4 open parentheses 0 close parentheses squared minus 36 end fraction equals 1 comma$ the y-intercept is

Putting this all together, here is the graph of f open parentheses x close parentheses.

Note that some coordinates are rounded to three decimal places.

A graph with an x-axis ranging from −8 to 8 and a y-axis ranging from −5 to 5. The graph contains three dashed lines, two vertical and one horizontal. The vertical dashed lines have equations x equals -3 and x equals 3, and the equation of the horizontal line is y equals 0. All dashed lines represent asymptotes. The graph contains three curves, which are separated by the vertical asymptotes. The curve on the left starts on the left side of the graph slightly below the x-axis, curves downward through the points (-6, -0.833) and (-4, -2.571), then continues to decrease along the left side of the dashed line x equals -3. The curve on the right starts at the bottom of the lower right quadrant, increasing along the right side of the dashed line x equals 3, curves upward through the points (4, 0) and (6, 0.167), then bends slightly downward to approach the x-axis from above. The middle curve resembles a U-shape with its vertex right of center, rising on both sides toward the asymptotes passing through the points (0, 1) and a minimum at (2, 0.9).

Notice the intercepts and asymptotes in the graph. The other points were obtained by substituting additional values of x into the function and evaluating. Since there are two vertical asymptotes, we need additional points to the left of x equals short dash 3 comma

between

and

and to the right of x equals 3.

hint

Graphing rational functions requires attention to detail, especially if the function has two or more vertical asymptotes. In the last example, notice that the graph on the interval between the vertical asymptotes has some unpredictable behavior. The only way to know for sure that there is a maximum or minimum point is to use calculus. For now, graphing utilities are very useful to explore characteristics of rational functions beyond their intercepts and asymptotes.

EXAMPLE

Consider the function $f open parentheses x close parentheses equals fraction numerator open parentheses x plus 1 close parentheses squared open parentheses x minus 3 close parentheses over denominator open parentheses x plus 3 close parentheses squared open parentheses x minus 2 close parentheses end fraction comma$ which has no common factor in its numerator and denominator.

Before examining its graph, note the domain is all real numbers except -3 and 2, or in interval notation, open parentheses short dash infinity comma space short dash 3 close parentheses union open parentheses short dash 3 comma space 2 close parentheses union open parentheses 2 comma space infinity close parentheses.

open parentheses short dash infinity comma space short dash 3 close parentheses union open parentheses short dash 3 comma space 2 close parentheses union open parentheses 2 comma space infinity close parentheses.

Consider the graph of f open parentheses x close parentheses.

A graph with an x-axis ranging from −8 to 8 and a y-axis ranging from −12 to 18. The graph contains two vertical dashed lines with equations x equals -3 and x equals 2, representing vertical asymptotes. There is a horizontal dashed line with equation y equals 1, representing a horizontal asymptote. The graph itself contains three curves, separated by the vertical asymptotes. To the left of the vertical asymptote, the graph starts on the left side of the upper left quadrant, slightly above the horizontal asymptote. As x increases, the graph increases, turning upward , increasing indefinitely along the left side of the dashed line x equals -3. Between the asymptotes, the curve resembles a U-shape with a flatter base, containing the points (-1, 0) and (0, 1 over 6). The graph increases sharply as it approaches the vertical asymptotes. The graph on the right contains the point (3, 0), increasing toward the horizontal asymptote to the right and decreasing toward the vertical asymptote to the left.

Note that the graph has the following characteristics. Compare these to the equation.

The vertical asymptote x equals short dash 3

corresponds to the factor open parentheses x plus 3 close parentheses squared

in the denominator. Notice that the graph rises indefinitely on both sides of the vertical asymptote x equals short dash 3.

The vertical asymptote x equals 2

corresponds to the factor x minus 2

in the denominator. Notice that the graph rises indefinitely on one side and falls indefinitely on the other side of the vertical asymptote.

The horizontal asymptote is y equals 1.

Note: in expanded form, $f open parentheses x close parentheses equals fraction numerator x cubed minus x squared minus 5 x minus 3 over denominator x cubed plus 4 x squared minus 3 x minus 18 end fraction.$

Since the degrees of the numerator and denominator are equal and the leading coefficients are both 1, the horizontal asymptote is y equals 1 over 1 equals 1.

The x-intercept open parentheses short dash 1 comma space 0 close parentheses

corresponds to the factor open parentheses x plus 1 close parentheses squared

in the numerator, which suggests that x equals short dash 1

has multiplicity 2.

Since f open parentheses 0 close parentheses equals 1 over 6 comma

the y-intercept is open parentheses 0 comma space 1 over 6 close parentheses.

The x-intercept open parentheses 3 comma space 0 close parentheses

corresponds to the factor open parentheses x minus 3 close parentheses

in the numerator, which suggests that x equals 3

has multiplicity 1.

Notice also that the graph of f open parentheses x close parentheses

crosses its horizontal asymptote twice.

watch

Check out this video to see an example of a graph of a rational function with a slant asymptote.

big idea

is a rational function in simplest form, then we know the following:

If is a factor of the numerator and k is even, then the graph touches the x-axis at
If is a factor of the numerator and k is odd, then the graph crosses the x-axis at
If is a factor of the denominator and k is even, then the graph will either increase or decrease indefinitely on both sides of the vertical asymptote
If is a factor of the denominator and k is odd, then the graph will increase indefinitely on one side and decrease indefinitely on the other side of the vertical asymptote

A graph can never cross its vertical asymptotes, but it can cross its horizontal, slant, or nonlinear asymptote.

To find the points where the graph of f open parentheses x close parentheses intersects with its horizontal asymptote, set equal to the value of y in the horizontal asymptote, then solve.

EXAMPLE

Consider the function $f open parentheses x close parentheses equals fraction numerator 2 x squared plus x over denominator x squared minus 9 end fraction comma$ which has horizontal asymptote y equals 2.

To determine if the graph of f open parentheses x close parentheses

crosses

set

f open parentheses x close parentheses equals 2

and solve:

$fraction numerator 2 x squared plus x over denominator x squared minus 9 end fraction equals 2$	Set
	Multiply both sides by
	Distribute.
	Subtract from both sides. This is now solved.

Since -18 is in the domain of f, the intersection point occurs at open parentheses short dash 18 comma space 2 close parentheses.

To check this, graph f open parentheses x close parentheses

along with its horizontal asymptote and note the intersection point.

To determine where the graph of a rational function intersects its slant or nonlinear asymptote, let’s first work an example before deciding a general strategy.

EXAMPLE

Consider the function $f open parentheses x close parentheses equals fraction numerator x to the power of 4 plus x squared plus x minus 1 over denominator x squared plus 4 end fraction.$

Since the degree of the numerator is two larger than the degree of the denominator, the graph of

has a nonlinear asymptote.

After performing long division, $f open parentheses x close parentheses equals x squared minus 3 plus fraction numerator x plus 11 over denominator x squared plus 4 end fraction comma$ which means that the equation of the nonlinear asymptote is y equals x squared minus 3.

Let’s now find all points where the graph of

intersects with its nonlinear asymptote.

$x squared minus 3 plus fraction numerator x plus 11 over denominator x squared plus 4 end fraction equals x squared minus 3$	Set $f open parentheses x close parentheses equals x squared minus 3 plus fraction numerator x plus 11 over denominator x squared plus 4 end fraction$ equal to
$fraction numerator x plus 11 over denominator x squared plus 4 end fraction equals 0$	Subtract from both sides and add 3 to both sides.
	Since multiply both sides by
	Solve for x.

Since

is in the domain of f open parentheses x close parentheses comma

the graphs intersect when x equals short dash 11.

The y-coordinate is $f open parentheses short dash 11 close parentheses equals open parentheses short dash 11 close parentheses squared minus 3 plus fraction numerator short dash 11 plus 11 over denominator open parentheses short dash 11 close parentheses squared plus 4 end fraction equals 118.$ Therefore, the point of intersection is

open parentheses short dash 11 comma space 118 close parentheses.

Notice that $fraction numerator x plus 11 over denominator x squared plus 4 end fraction$ is the remainder term of f open parentheses x close parentheses

after the long division was performed. In general, the x-coordinate of the point of intersection is found by setting the remainder term equal to 0.

step by step

To find any points where the graph f open parentheses x close parentheses

and its slant or nonlinear asymptote intersect…

Use long division to write in quotient-remainder form.
Set the remainder term equal to 0, then solve.
Evaluate at the value of x found in step 2.
Note that if long division produces a constant remainder, then the graph of doesn’t intersect with the asymptote.

try it

Consider the function $f open parentheses x close parentheses equals fraction numerator 2 x cubed minus 5 x squared minus 1 over denominator x squared minus 1 end fraction comma$ which has quotient-remainder form $f open parentheses x close parentheses equals 2 x minus 5 plus fraction numerator 2 x minus 6 over denominator x squared minus 1 end fraction.$

Find all points of intersection between the graph of f (x ) and its slant asymptote.

The equation of the slant asymptote is y equals 2 x minus 5.

To find the points where f open parentheses x close parentheses

and its slant asymptote intersect, set them equal to each other.

Before solving, note that the domain of f open parentheses x close parentheses

is any value of x for which the denominator is not equal to 0.

Since x squared minus 1 equals 0

when

the domain of f open parentheses x close parentheses

open parentheses short dash infinity comma space short dash 1 close parentheses union open parentheses short dash 1 comma space 1 close parentheses union open parentheses 1 comma space infinity close parentheses.

Now, solve the equation.

$2 x minus 5 plus fraction numerator 2 x minus 6 over denominator x squared minus 1 end fraction equals 2 x minus 5$	Set the quotient-remainder form of equal to the expression for the slant asymptote.
$fraction numerator 2 x minus 6 over denominator x squared minus 1 end fraction equals 0$	Subtract from both sides and add 5 to both sides.
	For the expression to be equal to 0, the numerator must be equal to 0.
	Add 6 to both sides.
	Divide both sides by 2. This is the solution.

Since 3 is in the domain of f open parentheses x close parentheses comma

it follows that the graph of f open parentheses x close parentheses

and its slant asymptote intersect when x equals 3.

To get the y-coordinate, substitute x equals 3

into the equation y equals 2 x minus 5 comma

since this equation is simpler.

We have y equals 2 open parentheses 3 close parentheses minus 5 equals 1.

This means that the intersection point is open parentheses 3 comma space 1 close parentheses.

Here is an example with a hole in the graph.

EXAMPLE

Consider the function $f open parentheses x close parentheses equals fraction numerator 2 x minus 10 over denominator x squared minus 25 end fraction comma$ which has factored form $f open parentheses x close parentheses equals fraction numerator 2 open parentheses x minus 5 close parentheses over denominator open parentheses x minus 5 close parentheses open parentheses x plus 5 close parentheses end fraction.$

First, note the domain of

is all real numbers except -5 and 5; or in interval notation, open parentheses short dash infinity comma space short dash 5 close parentheses union open parentheses short dash 5 comma space 5 close parentheses union open parentheses 5 comma space infinity close parentheses.

open parentheses short dash infinity comma space short dash 5 close parentheses union open parentheses short dash 5 comma space 5 close parentheses union open parentheses 5 comma space infinity close parentheses.

Notice the common factor of open parentheses x minus 5 close parentheses

between the numerator and denominator. This means that f open parentheses x close parentheses

can be written in a simpler form, namely $f open parentheses x close parentheses equals fraction numerator 2 over denominator x plus 5 end fraction comma$ where x not equal to 5.

Let $g open parentheses x close parentheses equals fraction numerator 2 over denominator x plus 5 end fraction.$

The denominator of is zero when which indicates that has vertical asymptote
The denominator is not zero when therefore, there is a hole in the graph of at or

Note that $f open parentheses 0 close parentheses equals fraction numerator 2 open parentheses 0 close parentheses minus 10 over denominator 0 squared minus 25 end fraction equals 2 over 5.$ This means that the y-intercept of the graph is open parentheses 0 comma space 2 over 5 close parentheses.

open parentheses 0 comma space 2 over 5 close parentheses.

Note that

is a factor of the numerator of f open parentheses x close parentheses comma

which would suggest that x equals 5

corresponds to an x-intercept of the graph of f open parentheses x close parentheses.

However, since x equals 5

is not in the domain of f, there is no x-intercept at open parentheses 5 comma space 0 close parentheses.

Since this is the only possibility, there is no x-intercept.

Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y equals 0.

Pulling all this together, here is the graph of f open parentheses x close parentheses.

A graph with an x-axis ranging from −10 to 6 and a y-axis ranging from −5 to 5. The graph has a horizontal dashed line overlapping the x-axis, representing a horizontal asymptote; and a vertical dashed line with equation x equals −5, representing a vertical asymptote. The graph also has two curves, where the left curve starts slightly below the x-axis on the left side of the lower left quadrant, runs close to the x-axis but decreases more sharply as it passes through (-6, -2), then continues downward along the left side of the vertical dashed line. On the right of the vertical dashed line, the graph starts very high in the upper left quadrant, decreases along the dashed line but curves over to pass through the points (0, 2 over 5), then levels out toward the x-axis in the first quadrant, also passing through an open circle at the point (5, 1 over 5).

Note that the point open parentheses short dash 6 comma space short dash 2 close parentheses

was added to show a point on the left side of the vertical asymptote.

IN CONTEXT

When using technology to graph a rational function that contains a hole, it is not usually visible on the graph. Most graphing programs actually make it look like the graph passes through the point where the hole is located. This is a shortcoming of the technology. This is why it is important to be able to analyze graphs using algebraic techniques—technology is not always perfect!

watch

In this video, we'll explore another example of a rational function whose graph contains a hole.

try it

Consider the function $f open parentheses x close parentheses equals fraction numerator 2 x minus 6 over denominator x squared minus x minus 12 end fraction.$

Find all vertical asymptotes of f (x ).

First, write f open parentheses x close parentheses

in factored form: $f open parentheses x close parentheses equals fraction numerator 2 open parentheses x minus 3 close parentheses over denominator open parentheses x minus 4 close parentheses open parentheses x plus 3 close parentheses end fraction$

The zeros of the denominator are x equals 4

and

This means that the domain of f open parentheses x close parentheses

is the set of all numbers except 4 and -3, expressed as open parentheses short dash infinity comma space short dash 3 close parentheses union open parentheses short dash 3 comma space 4 close parentheses union open parentheses 4 comma space infinity close parentheses

open parentheses short dash infinity comma space short dash 3 close parentheses union open parentheses short dash 3 comma space 4 close parentheses union open parentheses 4 comma space infinity close parentheses

using interval notation.

Since there is no common factor between the numerator and denominator, the zeros of the denominator correspond to vertical asymptotes.

The vertical asymptotes are x equals 4

and

Find the horizontal asymptote of f (x ).

Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote has equation y equals 0.

Find all holes in the graph of f (x ).

Since

has no common factor between the numerator and denominator, there is no hole in the graph of f open parentheses x close parentheses.

Find all x- and y-intercepts of the graph of f (x ).

Since

is already in simplest form, the x-intercept is found by setting the numerator equal to 0 and solving:

	Setting the numerator equal to 0.
	Add 6 to both sides.
	Divide both sides by 2. This is the answer.

The x-intercept is at the point open parentheses 3 comma space 0 close parentheses.

To find the y-intercept, find

This is possible since 0 is in the domain of f open parentheses x close parentheses.

$f open parentheses 0 close parentheses equals fraction numerator 2 open parentheses 0 close parentheses minus 6 over denominator 0 squared minus 0 minus 12 end fraction equals fraction numerator short dash 6 over denominator short dash 12 end fraction equals 1 half$

The y-intercept is located at open parentheses 0 comma space 1 half close parentheses.

Does the graph of f (x ) cross its horizontal asymptote?

The graph of f open parentheses x close parentheses

does intersect its horizontal asymptote, which has the equation y equals 0.

To find any such points, we would set $f open parentheses x close parentheses equals fraction numerator 2 x minus 6 over denominator x squared minus x minus 12 end fraction equals 0.$ We actually already found the solutions when finding the x-intercepts, which was the point open parentheses 3 comma space 0 close parentheses.

Thus, the graph of f open parentheses x close parentheses

intersects its horizontal asymptote at its x-intercept, open parentheses 3 comma space 0 close parentheses.

*Note: Graphing f open parentheses x close parentheses

would confirm all these results.

3. The Range of a Rational Function

Now that we are familiar with the graph of a rational function, we can examine the range of a rational function.

Recall that the range of a function is the set of all values that can be output values from a function.

EXAMPLE

Consider the function $f open parentheses x close parentheses equals fraction numerator 3 x minus 12 over denominator 2 x plus 4 end fraction comma$ as shown in the graph.

A graph with an x-axis ranging from −11 to 12 and a y-axis ranging from −10 to 12. The graph has a horizontal dashed line with equation y equals 1.5, representing a horizontal asymptote; and a vertical dashed line with equation x equals −2, representing a vertical asymptote. The graph also has two curves, where the left curve starts slightly above the horizontal asymptote on the left side of the upper left quadrant, then curves upward as x increases, passing through the points (-4, 6) and (-3, 10.5), then continues upward along the left side of the vertical dashed line. On the right of the vertical dashed line, the graph starts very low in the lower left quadrant, increases along the dashed line but curves over to pass through the points (0, -3), and (4, 0), then levels out toward the horizontal asymptote without crossing it.

We already know that the domain of this function is open parentheses short dash infinity comma space short dash 2 close parentheses union open parentheses short dash 2 comma space infinity close parentheses.

It appears that all values of y are possible except y equals 3 over 2 comma

which is also the horizontal asymptote.

This can be verified algebraically:

$fraction numerator 3 x minus 12 over denominator 2 x plus 4 end fraction equals 3 over 2$	Set
$fraction numerator 3 x minus 12 over denominator 2 open parentheses x plus 2 close parentheses end fraction equals 3 over 2$	Write denominator in factored form.
	Multiply both sides by
	Distribute.
	Subtract from both sides.

This equation has no solution. Therefore, there is no value of x for which y equals 3 over 2.

Thus,

is not in the range of f open parentheses x close parentheses.

The range of f open parentheses x close parentheses

open parentheses short dash infinity comma space 3 over 2 close parentheses union open parentheses 3 over 2 comma space infinity close parentheses.

EXAMPLE

We found earlier that the domain of f open parentheses x close parentheses

Notice that the graph intersects its horizontal asymptote y equals 1

twice.

In addition, all other real numbers are possible outputs of f open parentheses x close parentheses.

Conclusion: the range of f open parentheses x close parentheses

is all real numbers. In interval notation, this is written open parentheses short dash infinity comma space infinity close parentheses.

try it

Consider the function $f open parentheses x close parentheses equals fraction numerator 3 x over denominator x squared minus 4 x minus 12 end fraction.$

Determine the domain and range of f (x ).

First, write f open parentheses x close parentheses

in factored form: $f open parentheses x close parentheses equals fraction numerator 3 x over denominator open parentheses x minus 6 close parentheses open parentheses x plus 2 close parentheses end fraction$

As we can see, the denominator has zeros at x equals 6

and

This means the domain is all real numbers except x equals 6

and

or expressed using interval notation, open parentheses short dash infinity comma space short dash 2 close parentheses union open parentheses short dash 2 comma space 6 close parentheses union open parentheses 6 comma space infinity close parentheses.

open parentheses short dash infinity comma space short dash 2 close parentheses union open parentheses short dash 2 comma space 6 close parentheses union open parentheses 6 comma space infinity close parentheses.

To find the range, use the graph of f open parentheses x close parentheses comma

which is shown below.

A graph contains three curves. The leftmost curve begins in the third quadrant and runs along the negative x-axis, until turning downward to the right at (-3, -1), then continuing downward in the third quadrant almost vertically along the line x equals -2. The second curve begins high in the third quadrant just to the right of x equals -2, falls very quickly to the right until reaching the point (-1, 0.5), then falls more gradually, passes through (0.5, 0) and (4, -1), then falls sharply, continuing downward almost vertically on the left side of x equals 6. The rightmost curve falls quickly from high in the first quadrant slightly to the right of x equals 6, continues downward but turns and then runs horizontally toward the positive x-axis, but remaining in the first quadrant.

Looking at the graph, the horizontal asymptote is y equals 0.

The graph of f open parentheses x close parentheses

crosses this at open parentheses 0 comma space 0 close parentheses.

All other y-values are in the range of f open parentheses x close parentheses comma

which means the range is the set of all real numbers. Using interval notation, this is written open parentheses short dash infinity comma space infinity close parentheses.

Sometimes technology is very helpful in determining the range of a function.

EXAMPLE

Consider the function $f open parentheses x close parentheses equals fraction numerator 3 x squared over denominator x squared minus 4 x minus 12 end fraction.$ Its graph is shown below.

A graph with an x-axis ranging from −11 to 20 and a y-axis ranging from −6 to 15. The graph has a horizontal dashed line at y equals 3, representing a horizontal asymptote; and two vertical dashed lines at x equals −2 and x equals 6; representing vertical asymptotes. The graph also has three curves, separated by the vertical dashed lines. The left curve begins to the far left of the upper left quadrant, running along the underside of the horizontal asymptote; decreasing at first to the point (-6, -2.25), then increasing through the horizontal asymptote at (-3, 2), then continues to rise quickly along the left side of the vertical asymptote at x equals -2. The middle graph between the two vertical asymptotes, decreases and resembles an upside down U shape, but the right side falls more gradually than the left side does. The top of this curve is at (0, 0). The curve on the right starts high in the first quadrant slightly to the right of the dashed line x equals 6, descends quickly at first but becomes more gradual, passes through the point (13, 5), then continues to level off toward the horizontal dashed line without crossing it.

Note that there is a horizontal asymptote of y equals 3.

Note also the local minimum at open parentheses short dash 6 comma space 2.25 close parentheses

and the local maximum at open parentheses 0 comma space 0 close parentheses.

After carefully examining the graph, it looks like there is no point on the graph with outputs between 0 and 2.25, exclusively.

The range of this function is open parentheses short dash infinity comma space 0 close square brackets union open square brackets 2.25 comma space infinity close parentheses.

open parentheses short dash infinity comma space 0 close square brackets union open square brackets 2.25 comma space infinity close parentheses.

Note: without the graph of f open parentheses x close parentheses comma

calculus methods are required to locate the local minimum and maximum points.

summary

In this lesson, you learned that finding intercepts of a rational function f open parentheses x close parentheses

follows a similar procedure to finding intercepts of other functions, but more care needs to be taken so as not to violate the domain restrictions of f open parentheses x close parentheses.

Graphing rational functions requires knowledge of asymptotes, holes, and intercepts. Sometimes, however, more points are needed to understand the behavior of the function. Graphs can be used to find the range of a rational function. Graphing with technology can be very helpful when determining characteristics of a rational function but be aware of the shortcomings of technology (e.g., when using technology to graph a rational function that contains a hole, it is not usually visible on the graph). Therefore, it is important to be able to analyze graphs using algebraic techniques.

Source: THIS TUTORIAL HAS BEEN ADAPTED FROM OPENSTAX "PRECALCULUS". ACCESS FOR FREE AT OPENSTAX.ORG/DETAILS/BOOKS/PRECALCULUS-2E. LICENSE: CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL. Accessed by June 2022.

First, please create an account

Graphs of Rational Functions

Table of Contents

1. Intercepts of a Rational Function

2. The Graph of a Rational Function

3. The Range of a Rational Function