Rational Functions

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Objectives

• Define the domain and range of a function.
• Describe what is a rational function.
• Describe the features of the graph of a rational function: its asymptotes, zeros, and intercepts.
• Find the vertical and horizontal asymptotes of a rational function.
• Find the zeroes and intercepts of a rational function.
• Determine the domain and range of a rational function
• Graph rational functions.

Domain and Range

• Previously, we knew in Key Concepts of Functions that the domain is the set of all possible inputs of a function while the range is the set of all possible outputs of a function
  • But what does this exactly mean? Let’s study the function:

• We know that we can square any real number and it will give us a single result.
  - Therefore, the domain is the set of all real numbers.
  • However, know for the fact that when we square a number, it will always return a positive value.
• Therefore, the list of possible outputs, the range, is all positive real numbers.
• We can see that this is true by looking at the graph of .
  • The possible inputs along the -axis may span indefinitely, but the possible outputs along the -axis does not go below .

Definition
Domain and Range

The domain of a function is the set of all acceptable input values where is defined.

The range of a function is the set of all possible outputs of , .

Worked Problem
Finding the Domain and Range of a Function from a Table of Values

The table below shows the possible inputs and their corresponding outputs for the function . Find its domain and range.

Solution

The domain is the set of the all possible inputs, therefore we look at the left side of our table and list all inputs for :

The range is the set of all possible outputs, therefore we look at their corresponding outputs at the right side of our table, and list them:

Set and Interval Notation

• When we deal with domain and range, unlike functions defined as a table of values, some functions may have an with an infinite set of possible inputs and outputs.

• This is where we introduce the concept of set notation and interval notation.

  • We use these to indicate a range of values that might be included in a function’s domain or range.

• For instance, consider the graph of the function .

• From this graph, we can observe that the only possible inputs for this function ranges from to , including and themselves.

  • Therefore, we can express the domain of as all values over the interval .

• From observing this graph as well, the possible outputs of this function goes from to as well, including and themselves.

  • Therefore, we can express the range of as all values in the interval .

• The domain and range of a function are usually written as a set, and therefore we write them in set builder notation.

  • Therefore, the domain of can be formally defined as .
  • And the range of can be formally defined as .

Definition
Set Builder Notation

The set builder notation is a method of defining sets with infinite elements, and exist over an interval of values. They are often used to represent the domain and range of a function.

The domain of a function written in set notation can be expressed as:

The range of a function written in set notation can be expressed as:

or alternatively:

• To clear up the intricacies of this notation:
  • The or part indicates that the domain or range is in the set of real numbers
    • This means that the domain also includes both rational and irrational numbers as well.
  • The condition for or can either be an interval like or a restriction like .
    • We can also not include a condition or restriction to prove that a set includes all real numbers.
    • There can also be two or more non-overlapping conditions

Worked Example
Finding the Domain and Range of a Function from a Graph using Set Notation

Use the graph below to estimate the domain and range of .

Solution

The possible input values seems to extend indefinitely towards and , therefore, the domain of includes all real numbers.

We can express this as:

Notice that we don’t put any restrictions or conditions to show that it includes all real numbers

As for the range, the possible output vales seems to oscillate from to , therefore, the range of is:

• When we find the domain and range of a function, it is often assumed that we are always dealing with the set of real numbers.

  • Therefore, the or part in set notation becomes repetitive.
  • Often, the interval notation fixes this conundrum.

• The interval notation is written as follows:

  • The smallest number, , is written first, then followed by a comma.
  • Then the largest number, is written second, after the comma.
  • We use parentheses or to signify that the endpoint value is not included.
  • We use brackets or to signify that the endpoint value is included.

Description	Inequality/Restriction	Interval Notation	Graph
no restriction; all real numbers
is less than
is greater than
is less than or equal to
is greater than or equal to
is strictly between and , not including or
is between and , including
is between and , including
is between and , including and

Note

When writing interval notation with unbounded endpoints, that means intervals with endpoints at infinity, such as the ones below:

We always use a parenthesis as a number cannot ever be equal to or .

Worked Example
Expressing the Domain and Range of a Function from its Graph with Interval Notation

The graph of is displayed below.
Use the graph to estimate its domain and range. Express your answer in interval notation.

Solution

From its graph, we can say that it can accept any possible input.
This is because we can square any number, add and then take its positive square root.

Therefore, the domain of this function is:

However, the set of possible outputs does not go below and it extends indefinitely towards . This indicates that the range of the function is .

In interval notation, the range of this function is:

• Some functions may have domains or ranges that cannot be defined by a single interval.
  • In such cases, we use the union operator to combine two conditions or intervals.
  • For instance, the restriction can instead be expressed as and .
    • Therefore in interval notation, we can translate this to .

Description	Inequality/Restriction	Interval Notation	Graph
all real numbers, not including
all real numbers, not including and
is not in between and
is greater than , including , but is not equal to

What is a Rational Function?

• A rational function can be defined as a function that contains at least a single rational expression.
  • More generally, it is defined as a function that can be written as a quotient of two polynomial functions and .

Definition
Rational Function

A rational function is any function that can be represented as a quotient of two polynomial functions, and .

• This definition of a rational function allows us to distinguish other functions that is a quotient of two functions, but are not rational.
• For example, the functions below are not rational functions as the numerator and denominator is not made up of a polynomial function

• Some functions might not look like a rational function, but they can be written as one.

Domain of a Rational Function

• We now know from the previous lesson that it is not possible to divide by 0.
  • Therefore, when considering the domain of a rational function , our denominator must not equal to .
  • Therefore, by solving when the denominator equals zero, the solution set makes the function undefined.
    • We can then exclude these values to form our domain.

Worked Example
Finding the Domain of a Rational Function

Find the domain of the rational function .

Solution

First, we locate the denominator, which is .
Then, we find values for when is equal to .

Since the solution set is , we then exclude it to our domain.
Therefore, the domain of is: or in interval notation, .

Worked Example
Finding the Domain of a Rational Function with Factoring

Find the domain of the rational function .

Solution

To find the domain, we first retrieve the denominator .
Then, we find for values of where is .

Using zero product property, we can divide these into individual cases by equating when each of these factors equal zero. Recall that:

Zero multiplied by anything is zero, so if one factor is zero, then everything else must be zero

• For :

• For :

• For :

Therefore, the domain of is:

• in set notation or,
• in interval notation.

• In some cases, some algebraic manipulation is required.
  • This may be either adding or subtracting the rational expressions themselves, or using the quadratic formula.

Worked Example
Using the Quadratic Formula in Finding the Domain

Find the domain of the rational function .

Solution

First, we let the denominator equal to .

To solve this, we can use the quadratic formula.

Therefore, the domain is:

• in set notation,
• in interval notation.

• Not all rational functions have restrictions in their domain.
  • For instance, consider the function:

• This function is defined for all real numbers.
  • This is because there is no that will make equal to zero.
  • In fact, its graph does not even touch the -axis

General Instructions
Domain of a Rational Function

To find the domain of a rational function:

1. Set the denominator equal to zero.
2. Exclude the solution set in the domain.
3. Express your answer in any notation you prefer.

Range of a Rational Function

• As for the range, the process works for the same way, but we have to do some more work.
• For example, consider the function:

• First, we express the function as an equation for .
  • This can be done by replacing the function notation into .

• Then, we swap the and variables.

• Then, we solve for .
  • This can be done by grouping all terms in one side, then factoring them.

• This resulting function is the inverse of the function .
• By finding the domain of this inverse function, we get the range of our original function .
  • Getting the domain of the inverse function we get:

• Therefore, the range of is
  • in set notation.
  • in interval notation.

General Instructions
Finding the Range of the Rational Function

To find the range of the rational function :

1. Express the function as an equation by letting .
2. Then, find its inverse by swapping the and variables.
3. Solve the equation in terms of .
4. Find the domain of the resulting inverse function using methods in the previous section.

Worked Example
Finding the Range of a Rational Function

Find the range of the function .

Solution

First, let us express the rational function as a single rational expression.
We can do this by adding the two rational expressions using the methods discussed in Rational Equations

Then, we find its inverse.

Finding the domain of the inverse function, we see that:

Therefore, the range of is:

• in set notation,
• in interval notation.

Work in Progress!

The content beyond this section contains some missing information that might be essential in fully understanding the topic.

Warning

The content beyond this section assumes that you know:

• Domains of Radical Functions
• Complex Numbers
• Quadratic Inequalities

• Finding the range of a rational function is a cumbersome process.

  • This is because the process gets a little more difficult when we consider cases with a quadratic denominator or a higher degree denominator.

• For instance, consider the domain of the function:

• We can still apply the basic concepts of finding the range earlier.
  First, we express the function as an equation of , then we swap the variables.

• Then, we multiply both sides by whatever is in the denominator, .
  • We can then apply some algebraic manipulations

• Since we have a term, and we wanted to isolate the term, we can then use the quadratic formula.
  • To apply the formula, we must write our equation as

• We then now retrieve the coefficients for the quadratic formula.
  • Since we are solving for , we can say that and .
  • Since is the only term without a term then we set it as the constant term .
• Applying the quadratic formula, we get:

• By using the quadratic formula, we isolated in one side of the equation.
• Now we can analyze its domain.
  • The only way this function would be undefined is if:
    • The denominator is .
    • The radicand inside the square root becomes negative.
• Using the conditions we provided from its inverse, we now have the range restrictions of our original function .^range-restriction

• As for :

• As for , we apply the rules of quadratic inequalities

• This inequality will only satisfy when either or .
  Solving for :

• Solving for :

• Solving this inequality, we get that the range of our original function, , must be for all real numbers.
  • In set notation, it is
  • In interval notation, it is
• But what about the that we got from the original range restrictions?
  • It turns out that this is the horizontal asymptote of our original function.
  • This will be discussed in more detail in the next few sections.

Worked Example
Finding the Range of a Rational Function with a Quadratic Denominator

Find the range of the rational function

Solution

To find the range, we first express our function as an equation of .

Then, we proceed by finding its inverse.
First, we swap the and variables.

Next, we solve for .
Since the variable is a second degree term, we prepare to use the quadratic formula.
But first, we should rewrite it in the form .

We then apply the quadratic formula in terms of .
Since we are solving for , then the coefficients for , , and are:

Substituting the coefficients gives us:

We can treat this as the inverse of our original function .
Examining its domain should give us the range of .
This can be done by finding the values where the radicand is positive.

Applying the rules of quadratic inequalities:

• Since is a downward parabola, it can only be in the interval , where and and are the roots of the quadratic expression.

Therefore, the range of the original equation is:

• in set notation
• in interval notation.

• In some cases, the range of a function might be defined for all real numbers.
• Consider the example:

• The first step is to find the inverse of .
• We can do this by expressing the function as an equation of , swapping variables, manipulating the equation into , then applying the formula.
  • The resulting inverse function should give us:

• By solving for the domain of this function, we can find the range of our original function .
  Finding where the radicand :

• is an upward parabola.
  • The only way that this inequality is satisfied is if the smaller root and the larger root .
  • Using the quadratic formula to compute the roots, we can now begin to solve the inequalities:

• Since the restrictions are complex numbers (the radicand is negative in the square root sign), therefore, we can conclude here that the range of is the set of all real numbers.

Strategy
Finding the Range of a Rational Function

The strategy for finding the range might be a little different depending on the degree of the denominator.

If the degree of the denominator is linear:

• Express the function as an equation of .
• Find the inverse of by swapping the and variables.
• Solve for by multiplying both sides by the denominator.
• Rearrange so that all terms are in one side of the denominator.
• Factor the expression by .
• Then, divide both sides by the factored expression to find the inverse of and to isolate on one side of the equation.
• Find the domain of the resulting expression to get the range of .
• Express it in set or interval notation.

If the degree of the denominator is quadratic:

• Express the function as an equation of .
• Find the inverse of by swapping the and variables.
• Solve for by multiplying both sides by the denominator.
• Rearrange so that it takes the form .
• Apply the quadratic formula to isolate and to get the inverse of .
• Find the domain of the resulting expression to get the range of .
  • You may want to use rules of quadratic inequalities to find its domain.
  • You may need to use the quadratic formula again to find the roots.
  • If the restriction is a complex number, that means that the range is all real numbers.
• Express it in set or interval notation.

Zeroes of Rational Functions

• The zeroes of a function is a set of input values , where:

• In short, they are certain values where the function is equal to zero.
• However in a rational function, the only reasonable way where a function equals zero is when the numerator is equal to zero.
  • So, if a rational function can be expressed as:
    \

• If , then it follows that, .
  • Be ware that cannot be equal to zero.
  • Any value that makes is not included as a zero of a rational function, regardless if is also .

Definition
Zeros of Rational Functions

A zero of a rational function is any input value , where it satisfies the equation:

The zeros can be found by finding values where the numerator but does not result in a division by zero, or when the denominator .

• But what’s the difference between a zero and an -intercept?
  • The zero of a function is an actual numerical value, while an -intercept is an actual point in the coordinate plane that intersects the -axis.
  • One key idea is that if is a zero of a function , then is an -intercept of

Definition
-intercept of a Rational Function

An -intercept of a rational function is a point on the coordinate plane where the graph of crosses or intersects at the -axis, where the value is a zero of the rational function .

Worked Example
Zeros of Rational Functions

Find the zeroes of the rational function .

Solution

First, we solve when the numerator is equal to zero.

Solving the polynomial with zero product property, we get the values .
Notice that from our solution set, when we substitute to our denominator , we get:

makes the function undefined, so it does not belong as the zeros of
Therefore, the zeros of the rational function are
It also follows that the -intercepts of are and .

We can see this as we graph :

• It is also possible that a rational function does not have any zeros or -intercepts due to restrictions in their range.
  • For example, consider the function
  • Since the numerator is a constant , then it can never be equal to .
    • Therefore, it does not have an -intercept.
  • We can observe this by looking at its graph.

• Some rational functions may look like they have -intercepts but they do not.
  • Consider the function
  • At first glance, we might say that is a zero of .
  • Upon closer inspection, we can see that .
  • Therefore, this function does not have any -intercepts.
  • We can see that this is the case when we graph .

Strategy
Finding Zeroes and -intercepts of Rational Functions

If can be expressed as a rational function in the form:

Then, use the following strategy to find zeroes of rational functions.

• Solve when .
  • If cannot be equal to , then there are no zeroes or -intercepts as well.
• Check your answer by substituting the zeros back to the original function.
  • If then it is not a zero of the rational function.
• Express your answer as to find the -intercepts.

-intercept of a Rational Function

• The -intercept is the point at which the function crosses the -axis.
  • In other words, it is the value of the rational function at .

Definition
-intercept of a Rational Function

A -intercept of a rational function is the point where the function crosses the -axis. It can be found by simply evaluating .

Worked Example
-intercepts of Rational Functions

Find the -intercept of the rational function

Solution

To find the -intercept of a rational function, we simply evaluate .

Therefore, the -intercept is
We can see that this is the case when we graph the function itself.

• Note that not all rational functions have a -intercept.
  • Take for instance the function .
  • Since , therefore is undefined.
  • The graph shows us that there is no -intercept.

Asymptotes of Rational Functions

• An asymptote is a line that a function approaches as it gets closer and closer to a certain value.
• For instance, consider the graph of the function below:

• Here, we see two types of asymptotes: a vertical and a horizontal asymptote.

  • As the input gets closer to , the behavior of the graph gets closer to the vertical line .
  • At the same time, as the input gets closer to or , the value of the function approaches .

• We can see here that the graph of never crosses the lines and .

• There can be three types of asymptotes: vertical, horizontal, and oblique.

Vertical Asymptotes

• A vertical asymptote is a vertical line that the rational function never crosses.
• One intrinsic property of this asymptote is that as the input approaches the value , approaches or on either side of the asymptote

Definition
Vertical Asymptote

A vertical asymptote is a vertical line that the graph of a function never crosses.

For values near a vertical asymptote, the graph of a rational function tends to approach or on either side of the asymptote.

• In a rational function, vertical asymptotes only occur on input values that result in a division by zero.
  \

Definition
Vertical Asymptote of a Rational Function

Given a rational function expressed as a quotient of two polynomials:

A rational function has a vertical asymptote at if and only if it:

Worked Example
Finding Vertical Asymptotes in a Rational Function

Find the vertical asymptotes of the function

Solution

To find the vertical asymptotes of , we need to find the values where the denominator .

Using zero product property, we get the equations or .
Solving them gives us the vertical asymptotes of , namely and .

Substituting these values back to our original function, we see that it results to a division by zero:

.

These values are known to have removable discontinuities.
While they make the denominator zero, they do not result in a vertical asymptote.

• There are some cases where the expression in the denominator cannot be equal to .
  • These are the cases where a rational function might not have a vertical asymptote.
• For instance, consider the function .
  • Since cannot be equal to , therefore, it does not have a vertical asymptote.
  • This is evident when we take a look at its graph.

Worked Example
Finding Vertical Asymptotes in a Rational Function with Removable Discontinuities

Find the vertical asymptotes of .

Solution

First, we find where the denominator equals .

The only way that the denominator is zero is when .
Since does not have any roots, therefore, the equation does not have any real solutions.

Solving when gives us .
Substituting to our original function gives us:

The input value results in a , therefore, it has a removable discontinuity at that point.
Since there are no other possible factors that can equal zero, we can say that has no vertical asymptotes.

Analysis

Without solving and checking for the domain, we can immediately know that this function does not have any solutions.

First, we factor the numerator and the denominator.

Since we have an in both the numerator and the denominator, we can cancel it.
This eliminates any removable discontinuities in the function.

Checking the denominator, cannot be equal to zero as established and therefore, we can tell that does not have any vertical asymptotes.

Horizontal Asymptotes

• Just like vertical asymptotes, a horizontal asymptote is a horizontal line at which the graph of a rational function never crosses.
• Coincidentally, it is also the value that approaches as approaches or .

Definition
Horizontal Asymptote

A horizontal asymptote is a horizontal line at which the graph of a function approaches as approaches or .

Unlike the vertical asymptote, the function may or may not cross the horizontal asymptote.

• Suppose that the rational function .

• To find horizontal asymptotes, we use polynomial division.

  • When using polynomial division, we ensure that the degree of the numerator is greater than or equal to the degree of the denominator.
  • If the degree of the numerator, is less than the degree of the denominator , then we cannot divide and the horizontal asymptote is .

• If the degree of the numerator is equal to the degree of the denominator , the horizontal asymptote is the quotient of and .

Worked Example
Finding the Horizontal Asymptote of a Function 1

Find the horizontal asymptote of the function

Solution

Since the degree of the numerator, is and the degree of the denominator , we cannot divide and so the horizontal asymptote of is .

Worked Example
Finding the Horizontal Asymptote of a Function 2

Find the horizontal asymptote of the function .

Solution

Since the degree of the numerator , and the degree of the denominator , we can use polynomial division to find the horizontal asymptote.

Since the quotient is , the horizontal asymptote is .

• In general, we can only have a horizontal asymptote if the degree of the numerator is less than or equal to the degree of the denominator.
  • We can now generalize the conditions for a horizontal asymptote of a rational function.

Definition
Horizontal Asymptote of a Rational Function

Given a rational function that can be expressed as a quotient of two polynomials

A rational function has a horizontal asymptote if it satisfies these conditions:

• If , then the horizontal asymptote is .
• If , then the horizontal asymptote is

where is the degree of the polynomial function .

If , then there is no horizontal asymptote.

Oblique Asymptote

• An oblique asymptote is an asymptote that is neither a vertical line or a horizontal line.
• It is a straight line that a rational function approaches for larger values of
• For instance, consider the rational function:

• Using polynomial division, we get:

• Therefore, its oblique asymptote is .
• We can see that for larger values of , the rational function approaches the line .

Definition
Oblique Asymptote

An oblique asymptote is an asymptote that is neither a vertical line or a horizontal line.
It is a slanted line that a rational function approaches as approaches or .

An oblique asymptote only occurs if .
The quotient of the numerator and the denominator in polynomial division determines the equation of the oblique asymptote of the rational function.

Removable Discontinuities

• A removable discontinuity is a point in the graph of a function where is undefined.
• In other words, it is a “hole or a gap in the graph” where the function is undefined.
• For instance, consider the function:

• Observing the domain of the function from its denominator , this function must be undefined at and .
  • Therefore, we should expect an asymptote at the lines and .
• Upon graphing this function, we see that:

• Instead of an asymptote, we see a hole in the graph at .
  • This hole indicates that it is still undefined at that point.

Definition
Removable Discontinuities.

A removable discontinuity or a point discontinuity is a single point in the graph where the function is undefined.

It is different from a vertical asymptote that it does not approach or as the behavior of the graph approaches the point.

In a way, it is called “removable” as we can simply remove them and fill them so that the graph is continuous in that single point.

• One thing that we can do is to identify where a removable discontinuity might occur.
  • In our previous example, we can simply factor our denominator into .
  • We can now see an in both the numerator and the denominator.

• By cancelling this, we get the same exact graph, except without the hole in .

• This way, we can remove the discontinuity on the original function by simplifying it first, then evaluating the point discontinuity on our simplified expression.

Worked Example
Locating Point Discontinuities

Find any removable discontinuities in the rational function .

Solution

First, factor the numerator and the denominator.

We can see the factors , and are present in both the denominator and the numerator.

Equating them to will allow us to find where are they located.

• is our first point discontinuity.
• Solving gives : the second point discontinuity.
• Solving gives : the third point discontinuity.

Therefore, we have a discontinuity at .

Worked Example
Removing Point Discontinuities

Find and remove the point discontinuity on the function

Solution

First, we factor the denominator.
Since it is a quadratic with a leading coefficient that is not , we use factoring by grouping.

Substituting the factored expression, we get:

Since we have a in both the numerator and the denominator, then we have a removable discontinuity at the point where .
Solving for gives us : the location of our point discontinuity.

Going back to our original function, we can now cancel to get:

Finding should give us the point discontinuity

Therefore, the exact location of the discontinuity is at or at .