Watch this video lesson and get some practice reading and analyzing the graphs of rational functions. Learn how to identify asymptotes, domain, and range of these functions.
I actually think graphs of rational functions are pretty cool looking. What exactly is a rational function? I think of it simply as a fraction of polynomial functions. So, if you had two polynomials, those mathematical expressions made up of variables and their coefficients separated by pluses or minuses, and you combined them by placing one on top of another as a fraction, you would have a rational function. When you graph these somewhat crazy-looking functions, you get some really cool-looking graphs that do all kinds of things. Take a look at this one, for example. This is a graph of x^3 / (x + 1). Look at how it curves and does all kinds of weird things. It has two parts, too!
We can actually analyze this graph to find certain characteristics of this function. The characteristics that we will be talking about are the graph’s asymptotes, domain, and range. Let’s see what they are and how to find them.
There are actually two types of asymptotes we can look at here. First, a vertical asymptote is a vertical line that the graph will never touch or cross. You can find these easily by looking at the graph. Look for a straight up and down line that separates the graph. In our graph, we see that our graph is separated at the vertical line x = -1. That tells us that this is a vertical asymptote.
We have another type of asymptote called a horizontal asymptote. This asymptote tells us the behavior of the graph as x gets really big and really small. So, looking at the graph, we ask ourselves, what does the graph do at the far right and the far left? We see that as we get farther left and farther right, the graph goes higher and higher. So that means our horizontal asymptote is y = infinity.
We can also find these asymptotes by analyzing the function itself. To find our vertical asymptote, we set our denominator equal to 0 and solve it. In our case, our denominator is x + 1. Setting this equal to 0 and solving, we get x + 1 = 0 and x = -1.
To find our horizontal asymptote, we need to look at the exponent of the first term of our numerator and denominator. The exponent of our first term in our numerator is 3, and the exponent of the first term of our denominator is 1. Since 3 is larger than 1, then my horizontal asymptote is infinity, either positive or negative. If my exponent from the numerator is smaller than the denominator, then the horizontal asymptote will be y = 0. If the exponents are the same, then the horizontal asymptote is the fraction of the coefficients of these first terms. For example, if my function is (3x + 2) / (2x + 1), then my horizontal asymptote is y = 3/2.
The next characteristic is the domain, or valid inputs. The domain tells us all the x values that our function can be. To find these by looking at the graph, we look for x values that the graph never touches. Ask yourself, at what points does the graph jump? Well, the graph seems to jump or break apart at the vertical asymptotes. So all the vertical asymptotes of our function tell us what numbers are not part of our domain. For our function, our domain is all numbers except for -1.
To calculate this from the function, we do the same thing as we did to find the vertical asymptotes. We set the denominator equal to 0 and solve. We do this because we cannot have a function where we have division by 0. When we have division by 0, we have an invalid input, so by finding what x will give us division by 0, we find those values that our x cannot be.
The last characteristic is range, or possible outputs. These are all the values that our function or y can be. The range is usually unlimited unless there is a maximum or minimum that the function reaches. Our range will be limited by the horizontal asymptotes as well, if our function absolutely cannot equal that number. In our case, because our horizontal asymptote is infinity, our range is not limited, so it is the set of all numbers.
To calculate this from the function, we follow the same steps as we did to find our horizontal asymptotes by comparing the exponents of the first terms in both our numerator and denominator. We do not have to perform a check to see if our function absolutely cannot equal that number.
For example, the rational function 1 / x has a horizontal asymptote at y = 0 because the exponent of the first term in the denominator is higher than the exponent in the first term in the numerator. We can ask ourselves if there is an x that will give us y = 0. We know that the larger the number we divide 1 by, the smaller the number gets, but there really is no x that we can use to get y = 0 exactly. We get close to y = 0, very close, but we never reach it, even if we use a really, really, really big number for x.
What have we learned? We’ve learned that a rational function is a fraction of polynomial functions. The three characteristics that we looked at are asymptotes, domain, and range. We learned that a vertical asymptote is a vertical line that the graph will never touch or cross, while a horizontal asymptote is the behavior of the graph as x gets really big and really small. To find the vertical asymptotes, we look for vertical lines the graph jumps across or the values at which the denominator equals 0. For horizontal asymptotes, we look at what value the graphs is headed towards as x gets really big and really small.
To calculate it, we compare the exponents of the first terms in both the numerator and denominator. If the exponent of the first term of the numerator is larger than the exponent of the first term of the denominator, then our horizontal asymptote is y = infinity, either positive or negative. If the numerator’s exponent is smaller than the denominator’s exponents, then the horizontal asymptote is y = 0. If the exponents are the same, then the horizontal asymptote is the fraction of the coefficients of the first terms.
The domain of a function is the function’s possible inputs, or its possible x values. The domain is limited by the vertical asymptotes and any other values that x cannot be. The range of a function are its possible outputs, or possible y values. The range is limited by the horizontal asymptotes if the graph will never touch these.
The knowledge you gain from this lesson can be applied to help you do the following:
- Identify and calculate the vertical and horizontal asymptotes for a rational function
- Determine whether the range of a rational function is all possible values or limited
- Calculate the domain of a function