Video: Determining the Domain and Range of a Rational Function given Its Graph

Determine the domain and the range of the function 𝑓(𝑥) = (1/𝑥) − 5 in ℝ.


Video Transcript

Determine the domain and range of the function 𝑓 of 𝑥 equals one over 𝑥 minus five in the set of real numbers.

And then we’ve been given the graph of the function 𝑓 of 𝑥 equals one over 𝑥 minus five. And so, there are, in fact, two ways we can answer this question. Before we do, though, let’s begin by recalling what we actually mean by the domain and range of the function. The domain of the function is the set of all inputs, the set of all values of 𝑥, that will yield real outputs. And then the range is what we get when we input all of the domain.

Now, when trying to find a domain from an equation, there are a couple of things we need to be careful of. Firstly, if we have any square roots, we need to make sure that whatever is inside that square root is always positive. And secondly, we need to be really careful with rational expressions. With rational expressions, we need to ensure that our denominator is not equal to zero. We don’t want to be dividing by zero. So, in this case, we can see that we really don’t want 𝑥, which is the denominator of our fraction, to be equal to zero.

This means then that the domain, our input, the values of 𝑥 we’re going to choose, will be all real numbers. But we’re just not going to include zero. In set notation, we can say it’s the set of real numbers using this letter ℝ minus the set containing the number zero. But we can also deduce this from our graph. Our values of 𝑥 run from left to right. So, we look at our input left to right. We then see we have a vertical asymptote at 𝑥 equals zero. This is a vertical line where our values for our function get closer and closer to it, but never quite reach it. And so, we see that the domain is all possible numbers of 𝑥, not including 𝑥 equals zero.

So, what do we do about the range? Well, the range, we said, is the output after we’ve substituted in all of our domain. But another way to think about this is that when we find the inverse of a function, we switched the domain and range. So, we could actually find the domain of the inverse function. We can find the inverse function by rewriting our equation as 𝑥 equals one over 𝑦 minus five and then making 𝑦 the subject. Let’s add five to both sides, and then we’re going to multiply by 𝑦 and at the same time divide by 𝑥 plus five. And so, the inverse of our function is one over 𝑥 minus five.

Once again, to check the domain of our rational function, we’ll look at the denominator. Remember, we don’t want the denominator to be equal to zero. So, we want to find the value of 𝑥 such that 𝑥 plus five is not equal to zero. By subtracting five from both sides of our inequation, we see 𝑥 cannot be equal to negative five. And so, much as we denoted the domain, we can denote the range using set notation as the set of all real numbers minus the set including the number negative five.

If we want to think about this graphically, we look from top to bottom. We look in the 𝑦-direction. We see that there’s a horizontal asymptote at 𝑦 equals negative five. Once again, our function gets closer and closer to this but never quite touches it. So, we can see that graphically, our range is the set of all real numbers but it doesn’t include the value of 𝑦 as being equal to negative five. Either way, we’ve shown that the domain is the set of all real numbers minus the set including the number zero. And the range is the set of all real numbers minus the set including the number negative five.

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