Lesson Video: Domain and Range of a Rational Function Mathematics

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

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Video Transcript

Domain and Range of a Rational Function

In this video, we’ll go through several different methods to help us find the domain and range of a rational function. We’ll talk about the defining rule for finding the domain of a rational function. And we’ll talk about finding the range of a rational function either by using a sketch or by using algebraic techniques.

Before we start with this, let’s recall what we mean by the domain and range of a function. Let’s start with the domain of a function 𝑓 of 𝑥. The domain of 𝑓 of 𝑥 will be the set of inputs for our function. So one way of thinking about this is the value of 𝑥 we’re allowed to input into our function. For example, if our function 𝑓 of 𝑥 was the linear function two 𝑥 plus one, then we could say that our inputs 𝑥 could be any real value of 𝑥. And remember, we could write this in set notation 𝑥 is a member of the real numbers ℝ.

So we could have the domain of this function 𝑓 of 𝑥 as all real numbers. We could also pick a smaller set if we wanted. For example, we could say that we’re only allowed our inputs to be integers. We will write this as 𝑥 is a member of the integers ℤ. But this is just one possible example. What about if we were looking for the domain of the function 𝑓 of 𝑥 is equal to one over 𝑥?

Now we see something interesting. We can’t just input any real value of 𝑥 like we did last time. For example, if we had 𝑥 was equal to zero, then we would have 𝑓 of zero is one divided by zero. But we can input any other real value of 𝑥. So one possible domain for our reciprocal function would be the set of real numbers where we remove 𝑥 is equal to zero. And we write this as ℝ minus the set with just the element zero. And in fact, if we wanted to include as many real numbers in our domain as possible, this would be the biggest possible set because we can’t include zero.

Now let’s move on to the range of a function. One way of thinking about the range of a function is it’s the set of all possible outputs of our function. But of course, this will change depending on what we’re allowed to input into our function. In other words, this will depend on the domain of our function.

For example, let’s go back to our linear function 𝑓 of 𝑥 is two 𝑥 plus one, and let’s have the domain 𝑥 is any real number. Then we can sketch a graph of 𝑦 is equal to 𝑓 of 𝑥. We just get the straight line 𝑦 is equal to two 𝑥 plus one. We want to find the range of our function from this graph. Remember, the range of a function is the set of all possible outputs of our function. Well, the outputs of our function are the 𝑦-coordinates on the curve.

And if we pick any value of 𝑦, we can see in this case, there is a value of 𝑥 where 𝑓 of 𝑥 is equal to 𝑦. It’s important to remember the range of a function will depend on the domain of a function. For example, for 𝑓 of 𝑥 is two 𝑥 plus one, if we restrict our domain to be integers, then we’re only allowed to input integers to our function. So two times an integer plus one is also an integer, so we will only be able to output integers. Normally, however, when we look for the domain and range of a function, we want to include as many real values as possible.

The last thing we need to do before we talk about the range and domain of a rational function is recall what we mean by a rational function. We say if a function 𝑓 of 𝑥 is equal to 𝑝 of 𝑥 divided by 𝑞 of 𝑥 where 𝑝 of 𝑥 and 𝑞 of 𝑥 are both polynomials — in other words, it’s the quotient between two polynomials — then we call the function 𝑓 of 𝑥 a rational function. When we talk about the range and domain of a rational function, we’ll always be talking about this over the real numbers.

To find the domain of our rational function, we want to find as many values of 𝑥 over the real numbers where our function is defined. And the same is true for the range. Given as many inputs as possible into our rational function, we want to find as many possible outputs of our function. Let’s now take a look at some examples.

Determine the domain of the function 𝑓 of 𝑥 is equal to 𝑥 plus one divided by 𝑥 minus one.

The question is asking us to find the domain of a function 𝑓 of 𝑥. And we can see that 𝑓 of 𝑥 is the quotient of two polynomials. 𝑓 of 𝑥 is a rational function. Remember, the domain of a function is the set of possible inputs of that function. So let’s take a look at our function 𝑓 of 𝑥.

Well, 𝑓 of 𝑥 is the quotient of two polynomials. It’s 𝑥 plus one all divided by 𝑥 minus one. Let’s start by just looking at our numerator. Our numerator is 𝑥 plus one. Well, we know we can add one to any real number. So this won’t affect the domain of our function. Let’s now look at the denominator of our function. It’s 𝑥 minus one. Well, we can subtract one from any real number. So if we input a value of 𝑥, we’ll just get the quotient of two numbers.

But we have to be careful here. Remember, we can’t divide by zero. Division by zero is not defined. In other words, if we input 𝑥 in our denominator is equal to zero, our function won’t be defined. So we want to solve the denominator equal to zero. In other words, we want to know the values of 𝑥 where 𝑥 minus one is equal to zero. We see this is when 𝑥 is equal to one. So the only place where our function 𝑓 of 𝑥 is not defined is when 𝑥 is equal to one. And we can see this. If we input 𝑥 is equal to one into 𝑓 of 𝑥, we get one plus one divided by one minus one, which is equal to two divided by zero, which is of course not defined. But if we input any other value of 𝑥, our function will be defined. It will just give us the quotient of two numbers where the denominator is not equal to zero.

So in this case, we were able to show the domain of the function 𝑓 of 𝑥 is equal to 𝑥 plus one divided by 𝑥 minus one is all real numbers except when 𝑥 is equal to one.

But let’s think about what we’ve just shown about our function 𝑓 of 𝑥. In this case, we were able to find the domain of our rational function by only looking at where the denominator is equal to zero. But if we think about it, the exact same is true about any rational function.

If we have 𝑓 of 𝑥 is a rational function 𝑝 of 𝑥 divided by 𝑞 of 𝑥 where 𝑝 and 𝑞 are both polynomials, then we know that our polynomials 𝑝 and 𝑞 are defined for all real values of 𝑥. So the only time our function 𝑓 of 𝑥 won’t be defined is when our denominator 𝑞 of 𝑥 is equal to zero. This gives us a method of finding the domain for any rational function. We just need to find all the values of 𝑥 where our denominator is equal to zero.

Let’s go for an example of using this to find the domain of a rational function.

Determine the domain of the function 𝑓 of 𝑥 is equal to 𝑥 minus two divided by 𝑥 squared minus four.

The question wants us to find the domain of a function 𝑓 of 𝑥. We can see 𝑓 of 𝑥 is the quotient of two polynomials. This means our function 𝑓 of 𝑥 is rational. We know the domain of a function is the set of all possible inputs of that function. And what’s more, we know a trick for finding the domain of any rational function. We know if 𝑓 of 𝑥 is a rational function the polynomial 𝑝 of 𝑥 divided by the polynomial 𝑞 of 𝑥, then 𝑓 is only undefined when the denominator 𝑞 of 𝑥 is equal to zero. In our case, the polynomial in our denominator is 𝑥 squared minus four. So we need to find the values of 𝑥 where the denominator is equal to zero. We need to solve 𝑥 squared minus four is equal to zero.

The easiest way of doing this is to notice that 𝑥 squared is a square and four is a square, so this is a difference between squares. Remember, we can factor 𝑎 squared minus 𝑏 squared as 𝑎 minus 𝑏 times 𝑎 plus 𝑏. So we can factor 𝑥 squared minus four as 𝑥 minus two times 𝑥 plus two. So now we have the product of two factors is equal to zero. This means one of our factors must be equal to zero. Solving each factor is equal to zero, we get either 𝑥 is equal to two or 𝑥 is equal to negative two. And remember because our function 𝑓 of 𝑥 is rational, the only time it won’t be defined is when its denominator is equal to zero. In other words, we’ve shown the domain of 𝑓 of 𝑥 is all real numbers except when 𝑥 is equal to negative two or two.

There is one more thing worth pointing out in this example. We’ve said that two is not in the domain of our function. So let’s see what 𝑓 evaluated at two would give us. If we substitute 𝑥 is equal to two into our function 𝑓 of 𝑥, we get two minus two divided by two squared minus four, which simplifies to give us zero divided by zero. Remember, this is still not defined.

Let’s now do one more example of finding the domain of a rational function.

Determine the domain of the function 𝑓 of 𝑥 is equal to three divided by 𝑥 minus three plus one divided by 𝑥 plus four.

The question wants us to find the domain of the function 𝑓 of 𝑥. We recall the domain of a function is the set of possible inputs of our function. In our case, we can see that our function 𝑓 of 𝑥 is defined to be the sum of two functions. In fact, it’s the sum of two rational functions. Remember, we call a function rational if it’s the quotient between two polynomials. A linear function and a constant function are both examples of polynomials. So we need to find the set of possible inputs for our function, which is the sum of two rational functions.

Let’s start by recalling what we know about the domains of each of our rational functions individually. First, we recall any rational function will be defined everywhere except where its denominator is equal to zero. Remember, this is because we can input any value of 𝑥, and we’ll get a real number divided by another real number. But we can never divide by zero. It will always be undefined.

So let’s look at each term of 𝑓 of 𝑥 individually. Let’s start with three divided by 𝑥 minus three. We see this is a rational function. This will be defined everywhere except where its denominator is equal to zero. And we know its denominator is equal to zero only when 𝑥 is equal to three. We can do exactly the same with the second term one divided by 𝑥 plus four. This is also a rational function, so it will be defined everywhere except where its denominator is equal to zero, which in this case is when 𝑥 is equal to negative four.

But let’s think about what this means for our function 𝑓 of 𝑥. For example, if we were to substitute 𝑥 is equal to three into 𝑓 of 𝑥, we’d get 𝑓 of three is equal to three divided by three minus three plus one divided by three plus four. And if we simplify this, we get three over zero plus one over seven. Of course, three divided by zero is undefined. We can’t divide by zero. And of course, the same will be true when 𝑥 is equal to negative four. Since we’ve already explained one divided by 𝑥 plus four is not defined when 𝑥 is equal to negative four.

Every other input of 𝑥 will just give us the sum of two real numbers. So we’ve shown the domain of our function 𝑓 of 𝑥 is equal to three divided by 𝑥 minus three plus one divided by 𝑥 plus four is all real numbers except when 𝑥 is equal to negative four and when 𝑥 is equal to three.

Let’s now move on to an example of finding both the range and domain of a rational function.

Determine the domain and range of the function 𝑓 of 𝑥 is equal to one divided by 𝑥 minus five.

The question wants us to determine the domain and range of the function 𝑓 of 𝑥. And we can see that our function 𝑓 of 𝑥 is a rational function. It’s the quotient of two polynomials. We can also see we’re given a graph of the function 𝑓 of 𝑥. Let’s start by finding the domain of this function by using the graph. To start, remember, the domain is all the inputs of our function.

To find all the possible inputs of our function 𝑓 of 𝑥, let’s take a look at the values of 𝑥 our function can take. We want to find the values of 𝑥 where our function is undefined. We can see, for example, when 𝑥 is equal to six, our function outputs the value of one. So we can see six is in the domain of our function 𝑓 of 𝑥. In fact, there’s only one value where our function 𝑓 of 𝑥 is undefined.

If we draw the vertical line 𝑥 is equal to five, we can see that our function 𝑓 of 𝑥 does not intersect this line. This means our function is not defined when 𝑥 is equal to five. Every other vertical line will intersect our function, so this is the only point where our function is not defined. So we’ve shown our function 𝑓 of 𝑥 is defined everywhere except where 𝑥 is equal to five. In other words, the domain of 𝑓 of 𝑥 is the real numbers minus the point where 𝑥 is equal to five.

We now need to find the range of our function. Remember, the range of our function is the set of all possible outputs of our function. We can do something very similar to check whether a value is in the range of our function. For example, to check whether negative one is in the range of our function, we draw a horizontal line from negative one to the curve and then see the value of 𝑥 which gives us this output. We see from the graph when 𝑥 is equal to four, our function outputs negative one. Therefore, negative one is in the range of our function. We can see all horizontal lines will intercept our function except the one when 𝑦 is equal to zero. The line 𝑦 is equal to zero does not intercept our curve. In other words, no value of 𝑥 outputs the value of zero.

So for our function 𝑓 of 𝑥, there’s a value of 𝑥 which outputs every number except the value of zero. In other words, the range of 𝑓 of 𝑥 is the real numbers minus the point zero. Therefore, given a graph of the function 𝑓 of 𝑥 is equal to one divided by 𝑥 minus five, we were able to show the domain of this function is all real numbers except when 𝑥 is equal to five and the range of this function is all real numbers except for zero.

One important caveat of this example is that we can find the domain of our rational function by finding the vertical asymptotes and we can find the range of our function by finding the horizontal asymptotes.

Let’s now go through finding the domain and range of a rational function when we’re not given a graph of the function.

Determine the domain and range of the function 𝑓 of 𝑥 is equal to one divided by 𝑥 minus two.

The question is asking us to find the domain and range of the function 𝑓 of 𝑥. And we can see that our function 𝑓 of 𝑥 is a rational function. It’s the quotient of two polynomials. Remember, the domain of a function is the set of all possible inputs for that function and the range of a function is the set of possible outputs of that function. We also need to remember that any rational function will always be defined everywhere except when its denominator is equal to zero.

So for our rational function 𝑓 of 𝑥 to find the values of 𝑥 where our function is not defined, we just need to solve the denominator is equal to zero. This gives us 𝑥 minus two is equal to zero. And we can solve this. It gives us 𝑥 is equal to two. So we’ve shown our function 𝑓 of 𝑥 is defined everywhere except when 𝑥 is equal to two. In other words, the domain of 𝑓 of 𝑥 is all real numbers except when 𝑥 is equal to two.

But we still need to find the range of our function. There’s a few different ways of doing this. We’re going to do this by sketching a graph of our curve. And to do this, we should do the following two things. First, we need to check either side of every value of 𝑥 not in the domain of our function. And the only value of 𝑥 not in our function is equal to two. So let’s take a look at our function 𝑓 of 𝑥 around the value of 𝑥 is equal to two.

When our value of 𝑥 is a tiny bit less than two, our denominator will be a very small negative number. This tells us that our outputs will be very large but negative. So as 𝑥 gets closer to two from the left, our values of 𝑥 will be getting more and more negative. We can argue something similar to the right of 𝑥 is equal to two. This time, our denominator will be positive. So as 𝑥 approaches two from the right, our outputs will be getting larger and larger.

The next thing we want to check is what happens as our values of 𝑥 get larger and larger and what happens if they get more and more negative. Again, to do this, we can just look at our function 𝑓 of 𝑥. When 𝑥 is a very large positive number, our output will be one divided by a very large positive number. So what happens to our curve? Well, as 𝑥 is getting larger and larger, our outputs are getting closer and closer to zero. But they never reach zero, so we’ll just get closer and closer to the 𝑥-axis. The same is true as 𝑥 gets more and more negative. We’ll be dividing by a very large negative number, so our outputs will be getting closer and closer to zero.

So our function gets larger and larger as 𝑥 approaches two from the right and it gets more and more negative as 𝑥 approaches two from the left. However, no matter what input of 𝑥 we give, we can’t get our function to output zero. We can only get closer and closer to this output. So the range of our function 𝑓 of 𝑥 is all real numbers except for zero. Therefore, given the function 𝑓 of 𝑥 is equal to one divided by 𝑥 minus two, we were able to show the domain of this function is all real numbers except when 𝑥 is equal to two and the range of this function is all real numbers except for zero.

Let’s now go over the key points of this video. We were able to show that to find the domain of a rational function, we just need to find the values of 𝑥 which make the denominator zero, or even we just need to find the values of 𝑥 which make us divide by zero. And we have a few different ways of finding when a polynomial is equal to zero. For example, we know the factor theorem, the quadratic formula, and difference between squares.

Next, to find the range of a rational function, we just need to find all of the values that our function cannot output. We saw how to do this from a graph of our curve. However, if we’re not given this, we can also just sketch a graph ourself. We remember we also need to take particular care at what happens when 𝑥 is near the values where our function is undefined. We also need to consider what happens to our outputs as our inputs get larger and larger. We’ll also want to know what happens to our outputs as our values of 𝑥 get more and more negative. All of this can help us find the range of our rational function.

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