How To Find Domain And Range Of A Rational Function

In this explainer, we will learn how to detect the domain and range of a rational function either from its graph or its defining rule.

Before we start looking at how to find the domain and range of rational functions, let us remind ourselves what we hateful when nosotros talk about the domain and range of a part.

If we think of a function every bit a mapping that takes an input to an output, the domain would be the fix of inputs and the range the gear up of outputs. Consider the following mapping diagram:

We tin can see the inputs on the left and the outputs on the right. Here, the domain is the set up {1, 2, 3, 4} and the range is the set {two, four, 6, viii}. If we consider the function with domain {3, five, 7, nine}, then we can calculate the range by substituting each of the values from the domain into the function:

The range is, therefore, the set {11, 17, 23, 29}.

Earlier moving on, let us retrieve that

is the fix of natural numbers.
is the set of integers.
is the fix of rational numbers.
is the set of existent numbers.
is the set of complex numbers.

If we consider the function with domain (which means belongs to the set up of real numbers), it can be helpful when thinking nearly the range of the role to consider its graph.

Here, we can see the graph is a direct line, and every real number input has a real number output, and since the line continues infinitely in both directions, any real number output is possible. Therefore, the range is all the real numbers.

If we look at a quadratic function, for example, whose domain is the existent numbers, and if we look at the graph of , we can use this to determine the range.

Nosotros can encounter that for any input, the output is positive, and therefore the range of the function is any real number greater than or equal to cipher.

Now, given this recap, let us introduce the concept of rational functions. Generally, we tend to define the domain and range of functions over the existent numbers and we will do likewise here. We will take a different arroyo to working out the domains and ranges of rational functions, as it is not ever easy to sketch their graphs. Consider the function

Discover, for an input of we become .

Any division by zero is undefined, so nosotros have that the function is undefined at this betoken. Any nonzero input, nonetheless, will accept a corresponding output in the real numbers, and so nosotros can state that the domain is the real numbers excluding , written

By considering the nature of the part, nosotros can besides run into that whatever real number output can exist achieved with the exception of zilch: as gets progressively larger in magnitude, the output gets progressively smaller; withal, the output can never actually achieve zero. Therefore, the range of the role is

In general, to calculate the domain of a rational office, we demand to identify any betoken where the office is not defined, that is, whatsoever betoken that would give a denominator that is equal to zero. To find the range of a rational part, we need to identify any bespeak that cannot be achieved from whatsoever input; these can by and large be institute by considering the limits of the function every bit the magnitude of the inputs get very large. Let united states wait at some examples.

Instance 1: Finding the Domain and Range of a Rational Function with One Unknown in the Denominator

Discover the domain and range of the office .

Answer

From looking at the graph, information technology looks like the domain is and that the range is . However, nosotros should also check this algebraically.

Nosotros know that a rational function is non defined for any input that results in a zilch denominator. We can equate the denominator of the part to zero to find the undefined betoken. We take that which gives a solution of

This confirms that the domain is . To confirm the range we demand to identify all values that cannot be accomplished given the domain. As gets progressively larger in magnitude, the output tends to zip only will never really reach zero. Therefore, the range is indeed .

Allow united states now look at an instance where we are not given the graph and have to approach the question algebraically.

Example two: Finding the Domain and Range of a Rational Function Algebraically with 1 Unknown in the Denominator

Determine the domain and range of the role .

Answer

Call back that the expression is not defined, and from this we can decide that a rational role is non defined for any input that results in a zero denominator. We can equate the denominator of the function to zero to find the undefined betoken. We have that which gives a solution of

Therefore, nosotros can state the domain as . To discover the range, we need to identify all values that cannot be achieved given the domain. As gets progressively larger in magnitude, the output gets progressively close to nothing but volition never actually reach zero. Therefore, the range is .

At present, let us look at an case of finding the domain and range of a function with an unknown on the top and the bottom of the expression.

Example 3: Finding the Domain and Range of a Rational Part Algebraically with an Unknown in the Numerator and Denominator

Define a role on the real numbers by .

What is the domain of the function?
Find the one value that cannot have.
What is the range of the office?

Reply

Part 1

To detect the domain of the function, we demand to establish if there are values of for which is undefined. Equally this is a rational function, information technology will be undefined when its denominator takes a value of zero. Therefore, the graph of the office would have an asymptote when If nosotros subtract 5 from each side of the equation then dissever through by 4, we find that the asymptote has the equation . Therefore, the domain of the function is all real numbers except , notated .

Function 2

In lodge to determine the value that cannot take nosotros need to explore the limit of the function. That is, what happens equally gets large. To brand this procedure easier it is helpful to divide the top and bottom of the role by to get

From here, we tin come across that every bit gets progressively big, and get closer and closer to zero, and hence the function gets closer and closer to but never actually accomplish it.

Office 3

From the solution to function 2, we can come across that the range of the function is all existent numbers except , notated .

Let us now wait at a couple of more complicated examples. Firstly, a question where the rational function is presented as a sum of two functions, and, secondly, a rational function whose numerator and dominator are nonlinear.

Example four: Finding the Domain of a Sum of Rational Expressions

Make up one's mind the domain of the part .

Answer

Recall that rational functions are defined when their denominators are nonzero. From the part written in this form, we tin see that there are two points at which the function is undefined: when and when . This ways that the function is undefined when and . Therefore, the function's domain is all the real number except and three, notated .

As an additional piece of data, if nosotros were trying to find the range of this function, even though each of the rational expressions being summed cannot accept a value of zero themselves, there does be an input that is mapped to nil, which is . Therefore, the range of this function is actually the whole real numbers .

Example 5: Finding the Domain of a More Complicated Rational Expression

Find the domain of the real function .

Reply

Remember that a rational function is simply divers when its denominator is nonzero. Therefore, to observe the domain, we need to find the zeros of the equation To solve this, we tin can factor out an to get

From hither, we tin run into that we have a nothing when . The quadratic , however, has no real roots. Therefore, the only zip of the denominator is . The domain is, thus, all real numbers except 0, notated .

Fundamental Points

To observe the domain and range of rational functions recall the following steps:

To notice the domain of a rational function, we need to place any points that would pb to a denominator of zero.
To find the range of a rational role, nosotros need to identify all values that the office cannot take. Information technology can often exist helpful to look at the limits of the function to aid united states of america in this process.
It tin be helpful to consider the graph of the function to assist in the procedure of identifying the domain and range of a function.