So how do you find the domain of a function? So consider the function 2x-7. What is the domain of this function?
What is the list of all possible x values that can exist in this function? Whenever you have a linear function like the one that's listed, the domain is R-row numbers. So in interval notation, x can be anything. It can range from any value from negative infinity to positive infinity.
Likewise, if you have a quadratic function like x squared plus 3x minus 5, the domain is still R-row numbers. Or, if you have a polynomial function... such as 2x cubed minus 5x squared plus 7x minus 3, the domain is the same.
It's all real numbers. So if there are no fractions or square roots, if you just have a simple polynomial function, this is going to be the domain. Now what about if we have a rational function?
Let's say if we have a fraction. like 5 divided by x minus 2, how can we find the range, I mean not the range, but the domain of this function? In this function, x could be anything except a value that's going to produce a 0 in the denominator. So for instance, X minus 2 cannot equal 0. So therefore, X can't be positive 2. Because if you plug in 2, 2 minus 2 is 0. And whenever you have a 0 and the denominator is undefined, you can have a vertical asymptote. So for rational functions, set the denominator not equal to 0. And then you can find the value of X.
So how do you represent this using interval notation? So if we draw a number line, x could be anything except 2. So at 2, we're going to have an open circle. It can be greater than 2, or it can be less than 2. All the way to the left, you have negative infinity.
All the way to the right, positive infinity. So for the left side, x could be anything from negative infinity to 2, but not including 2. Or it could be anything from 2 to infinity. And so that's how you can write the domain.
use the interval notation for this example. Let's try another example. Let's say if we have 3x minus 8 divided by x squared minus 9x plus 20. So we have another rational function as seen by the fraction that we have. So what we need to do just like before, by the way you could try this problem if you want to. We need to set this not equal to 0. So x squared minus 9x plus 20 cannot equal 0. So how can we find the x values that will produce a 0 in the denominator?
What we need to do is we need to factor this trinomial. So what you want to do is you want to find two numbers that multiply to 20, but add to the middle coefficient, negative 9. So we know that 4 times 5 is 20, but they add up to 9, so we have to use negative 4 and negative 5, which still multiplies to a positive 20, but add up to negative 9. So therefore, x minus 4 times x minus 5 cannot equal 0. So we can say that x minus 4 cannot be 0, and x minus 5 cannot be 0. In the first one, let's add 4 to both sides. So x can't be 4, and for the second one, x can't be 5. Now, how do we represent this in interval notation? What I like to do is plot everything on a number line.
So if x can't equal 4, I'm going to put an open circle. And it can't equal 5 either. But it can be anything else. So now let's write the domain.
So from this section, it's from negative infinity to 4. But it does include 4. And then union, we have the second section. which goes from 4 to 5, and then union, the last section, which is 5 to infinity. So x could be anything except 4 and 5. Now what about this example? 2x minus 3, divided by x squared, plus 4. Go ahead and find the domain.
So let's begin by setting x squared plus 4 not equal to 0. So if we subtract both sides by 4, we'll get this. x squared cannot equal negative 4. Now, this will never happen. Whenever you square a number, you're going to get a positive number, not a negative number. For example, 3 times 3 is 9, negative 3 times negative 3 is positive 9. So, x squared will never equal negative 4. So, therefore, regardless of what x value you choose, the denominator will never be 0. If you plug in 2, your denominator will be 2 squared plus 4, which is 8. If you plug in negative 2, it's still going to be 8. If you plug in 0, it's going to be 4. It will never equal 0 in the denominator. So therefore, for this particular rational function, it's all real numbers.
The domain is from negative infinity to positive infinity. Now, what if you encounter a square root problem? So for example, what is the domain of the square root of x minus 4? How can we find the answer? Now for square roots, or any radical where the index number is even, you cannot have a negative number on the inside.
If it's odd, it could be anything, it's R-mole numbers. But for even radicals, or radicals with even index numbers, You have to set the inside greater than or equal to 0. It can't be negative. So for this one, all we need to do is add 4 to both sides.
So x is equal to or greater than 4. To represent that with a number line, we're going to have a closed circle this time. So it could be equal to or greater than, so we're going to shade to the right. So to the right we have positive infinity.
So the domain is going to be from 4 to infinity. Since it includes 4, we need to use a bracket in this case. Now, what about a problem that looks like this?
The square root of x squared plus 3x minus 28. How can we find the domain of this function? So just like before, we're going to set the inside of the square root function equal to or greater than 0. Now we need to factor. So let's find two numbers that multiply to negative 28, but that add to 3. So we have 7 and 4. Now, I need to add up to positive 3, so we're going to use positive 7 and negative 4. 7 plus negative 4 is positive 3, and 7 times negative 4 is negative 28. So, to factor, it's going to be x minus 4 times x plus 7. So, x can equal 4, and x can equal negative 7. Now, what I'm going to do is make a number line with these two values. Now, negative 7 and 4 are included, so let's put a closed circle.
Now, for this type of problem, we need to be careful. We need to find out which of these three regions will work. So, we need to check the sign.
We need to see which one is positive and which one is negative. So let's check this region first. If we pick a number that's greater than 4, like 5, and if we plug it into this expression, will it be positive or negative? Well if we plug in 5, 5 minus 4 is a positive number, and 5 plus 7 is a positive number.
When you multiply two positive numbers together, you're going to get a positive result. Now, if we pick a number between negative 7 and 4, let's say 0, and plug it in, 0 minus 4 is negative, 0 plus 7 is positive. A negative number times a positive number is a negative number. So if we choose any number in this region, it's going to give us a negative result.
Now if we choose a number that's less than negative 7, like negative 8, negative 8 minus 4 is negative, negative 8 plus 7 is negative. When you multiply two negative numbers, you're going to get a positive result. Now we can't have any negative numbers inside the square root symbol. So therefore... We're not going to have any solution in that region.
So therefore, we should only shade the positive regions. So now we can have the answer. So x...
can be less than negative 7, that's to the left, less than or equal to negative 7, or x can be equal to or greater than 4. Now to represent this using interval notation, it's going to be from negative infinity to negative 7, and then union, we're going to start back up at 4 to infinity. And we need to use brackets at 7, I mean negative 7 and 4. because it includes those two points. We have a closed circle there.
So that's how you can find the domain of this type of function. Now sometimes, you may have a fraction with a square root. So what do you do if the square root is in the denominator of the fraction? Now, If the square root was not in the denominator, we would set the inside equal to and greater than 0. But, we can't have a 0 in the bottom of a fraction, so this time, we can only set the inside just greater than 0. So, x has to be greater than negative 3. So, the domain is simply going to be from negative 3 to infinity, but not including negative 3. Now, let's consider another example. So we're going to have a fraction again, but with a square root in the numerator.
What do you think the domain for this function is going to be? Now, if you have a square root in the numerator, you need to set the inside equal to or greater than 0. So x is equal to and greater than 4. Now we know that in a denominator we can't have a 0. So we're going to set it equal or not equal to 0. And we could factor it. So this is going to be x plus 5 times x minus 5, using the difference of squares method.
So x cannot equal negative 5, and it can't equal 5. So now let's make a number line. So we have negative 5, 4, and 5. Thank you. So we're going to have an open circle at negative 5 and 5. And then x is equal to or greater than 4. So we're going to have a closed circle at 4 and shade to the right. So there's nothing really to write here because x is not going to equal to anything less than 4. It equals everything greater than 4 included 4, but just not 5. So how do we represent that in interval notation?
So this is the first part. So we're going to start with 4 using brackets and stop at 5 using parentheses, since it does not include 5. And then union for the second part is going to go from 5 to infinity. So that's how you can represent the answer using interval notation.
Now what would you do if you have a fraction that contains a square root in the numerator and also in the denominator? Try this. So let's focus on the numerator. We know that x plus 3 is equal to or greater than 0, which means x is greater than or equal to negative 3. So if we plot that on our number line... this is what we're going to have so it's from negative 3 to infinity now let's focus on the square root in the bottom so we know that x squared minus 16 has to be only greater than 0 but not equal to it because if it's on the bottom it can't be 0 So if you have a square root on the top, you set it equal to and greater than 0. If it's on the bottom, simply just greater than 0. So what we need to do first is factor this expression.
It's going to be x plus 4 and x minus 4. So x can't be negative 4 and x can't be 4. But it can be equal to values in between. So we're going to make a second number line. Now the reason why I can't equal it is because we don't have the underlined symbol. It's only greater than 0, but not equal to 0. So let's start with an open circle at negative 4 and 4. Now whenever you have like two circles on a number line due to a square root function, I like to do a sign test to find out which regions it's going to be negative. In this example, it's going to be positive above negative 3, but negative below negative 3. Now, let's plug in some numbers.
So, if we plug in a 5 to check the region on the right, 5 plus 4, using this expression, that's going to be positive, and 5 minus 4 is positive. So, two positive numbers multiply to each other. will give us a positive result.
If we plug in 0, 0 plus 4 is positive, 0 minus 4 is negative. So positive times a negative number is a negative number. And if we plug in negative 5 to check that region, negative 5 plus 4 is negative, negative 5 minus 4 is still negative. Two negative numbers will multiply and give you a positive result.
So now what should we do at this point? Now we know that we can't have any negative numbers inside a square root symbol. So it's not going to be anything between negative 4 and a 4. So for the square root on the bottom, x can be greater than 4, and it can be less than negative 4, but nothing in between.
So now, what we need to do is find the intersection of these two number lines. We've got to find out where it's true for both functions. So I'm going to create a hybrid number line.
So I'm going to put negative 4, negative 3, 4, and infinity. And negative infinity as well. So looking at the first one, it's not going to work if we have anything that's less than negative 3. So therefore, we should have nothing on the left side.
So this is going to be irrelevant. Because it's true for the second part, but it doesn't work for the first one. Now, we're not going to have anything between negative 3 and 4, because this is an empty region between negative 3 and 4. Even though it works for this one, it doesn't work for the second one.
So therefore, the answer has to be from 4 to infinity. This region is true for both number lines. This region here applies to this number line, and also this one as well.
Because somewhere between negative 3 and infinity, there's a 4. Now it has to be an open circle, not a closed circle. So 4 to infinity overlaps for this function on top, the square root on top, and also the square root on the bottom. So that's going to be the answer. The domain is going to be 4 to infinity.
So if you have two square root functions in a fraction, you need to... make two number lines separately, and find a region of intersection where it's true for both number lines. And so, in this example, that's from 4 to infinity.
And so that's how you do it. So now you know how to find the domain of a function, such as linear functions, polynomial functions, rational functions, and also square root functions.