How To Find The Domain of a Function - Radicals, Fractions & Square Roots - Interval Notation

So how do you

find

the

domain

of a

function

? Consider the

function

2x minus 7. What is the

domain

of this function? What is the list of all possible values ​​of x that can exist in this function as long as it has a linear function like the one listed? the domain is all real numbers, so in

interval

notation

x can be anything that can vary from any value from negative infinity to positive infinity. Likewise, if you have a quadratic function like x

square

d plus 3x minus five, the domain is still real numbers, or if you have a polynomial function like 2x cubed minus 5x

square

d plus 7x minus 3. the domain is the same, they are all numbers. reals, so if there are no

fractions

or square

roots

, if you just have a simple polynomial function, this will be the domain.

And now that? about whether 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'm not talking about the range, but the domain of this function in this function x could be anything except a value that is going to produce a zero in the denominator, so for example x minus two cannot be equal to zero , therefore 0 and then you could find the value of x, so how is this represented using

interval

notation

? So if we draw a number line, x could be anything except two, so at 2 We're going to have an open circle, it can be greater than 2 or it can be less than 2.

From the left you have negative infinity 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 this is how you can write the domain using 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. Then we have another rational function as seen in the fraction we have, so what we have to do is as before. By the way, you can try this problem if you want.

We need to set this to nonzero, so x squared minus 9x plus 20 cannot equal zero, so how can we find the values ​​of x that will produce a zero in the denominator? What we need to do is factor this trinomial, so what we want to do is find two numbers that multiply to 20 but we add to the middle coefficient minus nine, so we know that four times five is twenty, but they add up to nine, so we have to use negative four and negative five, which still multiply to plus twenty, but add up to negative nine, therefore x minus four times x minus five cannot be equal to zero so we could say that x minus four cannot be zero and x minus five cannot. can be zero in the first let's add four to both sides so that x cannot be four and for the second x cannot be five 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 any 5, but it can be anything else, so now let's write the domain, so from this section it's from negative infinity to four, but it includes four and then the union we have the second section which goes from four to five and then the union of the last section which is from 5 to infinity, so x could be anything except 4 and 5. Now, what about this example, two x minus three divided by x squared plus four?

Go ahead and find the domain so let's start by stating that x squared plus four is not equal to zero so if we subtract both sides by four we will get this x squared cannot equal negative four now this will never happen every time If you square a number you will get a positive number, not a negative number, for example three times three is nine minus three times minus three is plus nine, so x squared will never equal negative four, so regardless Whatever value of x you choose, the denominator will never be zero if you connect two, your denominator will be two squared plus four, which is eight if you connect minus two. it will still be eight if you enter zero it will be four it will never be equal to zero in the denominator therefore for this particular rational function they are all real numbers the domain is from negative infinity to positive infinity what happens if you encounter a problem of square root, so, for example, what is the domain of the square root of x minus 4?

How can we now find the answer for square

roots

or any radical where the index number is even? It cannot have a negative number inside if it is odd. It could be anything, they are armature numbers, but for even

radicals

or even index number

radicals

you have to set the inside greater than or equal to zero, it can't be negative, so for this one all we have to do is add four to both sides, so x is equal to or greater than four to represent that with a number line this time we are going to have a closed circle, so it could be equal to or greater than, so we are going to shade to the right so that towards right has positive infinity, so the domain will be from 4 to infinity, since it includes 4, we need to use a square bracket in this case.

Now what about a problem that looks like this? The square root of x squared plus three x minus twenty eight, how can we find the domain of this function, so, as before, we will set the interior of the square root function equal to or greater than zero, now we need to factor, like this Let's find two numbers that multiply to negative 28 but add three, so we have seven and four now I need to add positive 3, so we will use positive 7 and negative 4. 7 plus negative 4 is positive 3 and 7 times negative 4 is negative 28. So it's a factor that is going to be x minus 4 times x plus 7. so x can be equal to 4 and x can be equal to minus 7.

Now what I'm going to do is make a number line with these two values, now they are included minus 7 and 4, so let's put a number line circle now for this type of problem, we have to be careful, we have to find out which of these three regions will work, so we have to check the signs, we have to see which one is positive and which one is negative, so Let's review this region. First, if we choose a number greater than four, like five, and if we plug it into this expression, will it be positive or negative?

Well, if we connect five, five minus four is a positive number and five plus seven is a positive number when you multiply. two positive numbers together you will get a positive result now if we choose a number between negative seven and four, let's say zero and we replace it with zero minus four is negative zero plus seven is positive a negative number multiplied by a positive number is a negative number so if we choose any number in this region will give us a negative result now if we choose a number that is less than negative seven like negative eight minus eight minus four is negative negative eight plus seven is negative when you multiply two negative numbers it will get a positive result now we can't have no negative number inside the square root symbol, therefore we are not going to have any solution in that region, so we only need to shade the positive regions, so now you can have the answer then x can be less than negative seven that's on the left less than or equal to negative seven or use brackets on seven, I mean minus seven and four, because it includes those two points, we have a closed circle there, that's how you can find the domain of this type of function now sometimes you can 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 were not in the denominator, we would set the inside equal and greater. than zero but we can't have a zero at the bottom of a fraction so this time we can just set the inside a little bit greater than zero so x has to be greater than negative three so the domain will simply be from negative three to infinity but not including negative three. 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 of this function will be now if you have a square root in the numerator you need to set the interior equal to or greater than zero so that x is equal to and greater than four.

Now we know that in a denominator we cannot have a zero, so we are going to set it equal or not equal to zero now we can factor it, so it will be x plus five times x minus five using the difference of squares method, so x It can't be equal to negative five and it can't be equal to five, so now let's make a number line to have negative five. four and five, so we'll have an open circle at negative five and five and then x is equal to or greater than four, so we'll grab four more closely and shade to the right, so there's really nothing to write here because x doesn't is going to be equal to anything less than 4. is equal to everything greater than 4, including 4, but not 5.

So how do we represent that in interval notation? This is the first part, so we will start with four using parentheses and stop at five using parentheses since it does not include five and then the union for the second part will go from five to infinity, this is 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 three is equal to or greater than 0, which means that 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 and the bottom so we know that x squared minus 16 only has to be greater than zero, but not equal. because if it's at the bottom it can't be zero, so if you have a square root at the top, you set it equal to and greater than zero, if it's at the bottom, it's simply greater than zero, so what we need to do first is factoring. This expression will be x plus 4 and x minus 4. so x cannot be negative 4 and x cannot be 4. but it can be equal to values ​​in between, so let's make a second number line. now the reason I can't equal this is because we don't have the underlying symbol, it's just greater than 0 but not equal to 0. so let's start with an open circle at negative four and four now whenever you have two circles on a line numerical due to a square root function.

I like to do a breast test to find out which regions will be negative. In this example, it will be positive above negative 3 but negative below negative 3. Now let's plug in some numbers, so if we enter a 5 to check the region to the right 5 plus 4 using this expression, that will be positive and 5 minus 4 is positive, so two positive numbers multiplied together will give us a positive result if we enter zero zero plus four is positive zero minus four is negative, so a positive number multiplied by a negative number is a negative number and if we plug in negative 5 to verify that the region negative 5 plus 4 is negative negative 5 minus 4 is still negative , two negative numbers will be multiplied and we give a positive result, so what should we do at this point?

Now we know that we can't have negative numbers inside a square symbol, so there will be nothing between negative four and four, for the square root at the bottom. x could be greater than four and it could be less than negative four, but nothing in between, so now what we need to do is find the intersection of these two number lines. We have to figure out where is 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 won't work if we have something that is less than negative 3.

So we shouldn't have anything on the left side so this will be irrelevant because it's true for the second part but it doesn't work for the first now we're not going to have anything between negative 3 and 4 because This is an empty region between negative 3 and 4. Although it works for this one, it doesn't work for the second one, 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 because somewhere between negative 3 and infinity there is 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 at the top and also the square root. at the bottom, that will be the answer, the domain will be from 4 to infinity, so if you have two square root functions on a fraction, you have to make two separate number lines and find a region of intersection where it is true for both numbers. lines and in this example that is 4 toinfinity and this is 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.