Find the Domain and Range: Rational Functions (with Denominators)
Find the Domain and Range: Functions with Denominators - How it Works - Video
Horizontal Asymptote Chart
Horizontal Asymptote Chart:
This is how we can quickly tell if a function has any horizontal asymptotes.
If the top exponent is less than the bottom exponent, then our horizontal asymptote is at y =0.
If the top exponent is equal to than the bottom exponent, then our horizontal asymptote is at the division of our coefficients. In the case of the picture, our horizontal asymptote is at y =5/2.
If the top exponent is greater than the bottom exponent, then we do not have a horizontal asymptote.
Example 1
Example 1:
Here we f(x) = x2 / (x2 - 9). First things first we are going to graph it. Now if you don't have a calculator that is okay, we can still solve for the domain without it, but sometimes the range can be a bit trickier, but it is easier to see what is happening with a graph.
Since the denominator has a quadratic, ax2 + bx + c, where the b coefficient is 0, we can solve for x. So let's set the denominator equal to zero => x2 - 9 = 0. Now let's solve for x, x2 - 9 ==> x2 = 9 ==> x = ±3. We also have a difference of squares so you can have factored the quadratic as well.
Remember we can't divide by 0. These numbers, -3 and 3, are the only numbers in this function that would make the denominator zero. This means when it happens it creates vertical asymptotes. So we have vertical asymptotes at x = -3 and x = 3.
With this information we can create our domain. We have a few ways to write our domain. We can use a set with curly brackets or just inequalities or interval notation. Let's write some of the different ways: {x ϵ ℝ, x ≠ -3, 3} and (-∞, -3) U (-3, 3) U (3,+∞). If we look at the graph, it follows what we just wrote down.
Now, let's find our range. This is where it can get tricky. We have to look at the exponents in the numerator and denominator. Since we have the same, we need to divided the coefficients of each one. 1 divided by 1 is 1 so our horizontal asymptote is y = 1. If we look at the graph we notice that there aren't any values between 0 and 1. There isn't really a way to find that out without guessing and checking if you don't have a calculator.
Like our domain, we have a few ways to write our range. Let's write some of the different ways: {y ϵ ℝ, y ≤ 0 or y > 1} and (-∞, 0] U (1,+∞). If we look at the graph, it follows what we just wrote down.
Example 2
Example 2:
Here we f(x) = (x -2) / sqrt(x + 4). First things first we are going to graph it. Now if you don't have a calculator that is okay, we can still solve for the domain without it, but sometimes the range can be a bit trickier, but it is easier to see what is happening with a graph.
So let's set the denominator equal to zero. Since we have a square root, we can set just the inside equal to zero because we can't negative input value in square roots. So we have x + 4 = 0 => x = -4.
Remember we can't divide by 0. This number, -4, is the only number in this function that would make the denominator zero. This means when it happens it creates a vertical asymptote. So we have a vertical asymptote at x = 4.
With this information we can create our domain. We have a few ways to write our domain. We can use a set with curly brackets or just inequalities or interval notation. Let's write some of the different ways: {x ϵ ℝ, x > -4} and (-4,+∞). If we look at the graph, it follows what we just wrote down.
Now, let's find our range. This is where it can get tricky. We have to look at the exponents in the numerator and denominator. Since we have the top, 1, is greater than the bottom, 1/2, we don't have any horizontal asymptotes.
Like our domain, we have a few ways to write our range. Let's write some of the different ways: {y ϵ ℝ} and (-∞,+∞). If we look at the graph, it follows what we just wrote down.
Example 3
Example 3:
Here we f(x) = (x -1) / (x3 - 4x). First things first we are going to graph it. Now if you don't have a calculator that is okay, we can still solve for the domain without it, but sometimes the range can be a bit trickier, but it is easier to see what is happening with a graph.
In the denominator, we have a cubic function x3 - 4x. We can factor out an x from each term => x * (x2 - 4). We can now set the first part and the second part equal to zero. Now we have x = 0 and (x2 - 4). We, now, have a quadratic, ax2 + bx + c, where the b coefficient is 0. Now let's solve for x, x2 - 4 ==> x2 = 4 ==> x = ±2.
We also have a difference of squares so you can have factored the quadratic as well, which was what we did in the picture.
Remember we can't divide by 0. These numbers, 0, -2 and 2, are the only numbers in this function that would make the denominator zero. This means when it happens it creates vertical asymptotes. So we have vertical asymptotes at x = 0, x = -2, and x = 2.
With this information we can create our domain. We have a few ways to write our domain. We can use a set with curly brackets or just inequalities or interval notation. Let's write some of the different ways: {x ϵ ℝ, x ≠ 0, -2, 2} and (-∞, -2) U (-2, 0) U (0, 2) U (2,+∞). If we look at the graph, it follows what we just wrote down.
Now, let's find our range. This is where it can get tricky. We have to look at the exponents in the numerator and denominator. Since top is less than the bottom, we have a horizontal asymptote at y = 0. If we look at the graph we have 4 different graphs, that would be to figure out without guessing and checking a few points.
Like our domain, we have a few ways to write our range. Let's write some of the different ways: {y ϵ ℝ} and (-∞,+∞). If we look at the graph, it follows what we just wrote down.