so how do you find the domain of a function so consider the function 2x minus 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 all real numbers so in interval notation x can be anything it could range from any value from negative infinity to positive infinity likewise if you have a quadratic function like x squared plus 3x minus five the domain is still real 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 zero in the denominator so for instance x minus two cannot equal zero so therefore x can't be positive two 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 could 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 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. 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 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. 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 zero so x squared minus 9x plus 20 cannot equal zero so how can we find the x values that will produce a zero 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 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 multiplies to a positive twenty but add up to negative nine so therefore x minus four times x minus five cannot equal zero so we could say that x minus four cannot be zero and x minus five cannot be zero in the first one let's add four to both sides so x can't be four and for the second one x can't 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 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 four but it does include four and then union we have the second section which goes from four to five 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 two x minus three divided by x squared plus four go ahead and find the domain so let's begin by setting x squared plus four not equal to zero so if we subtract both sides by four we'll get this x squared cannot equal negative four now this will never happen whenever you square a number you're going to get a positive number not a negative number for example three times three is nine negative three times negative three is positive nine so x squared will never equal negative four so therefore regardless of what x value you choose the denominator will never be zero if you plug in two your denominator will be two squared plus four which is eight if you plug in negative two is still going to be eight if you plug in zero is gonna be four it will never equals um zero 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 armor numbers but for even radicals or radicals of even index numbers you have to set the inside greater than or equal to zero it can't be negative so for this one all we need to do is add four to both sides so x is equal to or greater than four 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 three x minus twenty eight how can we find the domain of this function so just like before we're gonna set the inside of the square root function equal to or greater than zero now we need to factor so let's find two numbers that multiply to negative 28 but that add to three so we have seven and four 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 it's a 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 signs we need to see which one is positive and which one is negative so for let's check this region first if we pick a number that's greater than four like five and if we plug it into this expression will it be positive or negative well if we plug in five five minus four is a positive number and five plus seven 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 seven and four let's say zero and plug it in zero minus four is negative zero plus seven 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 seven like negative eight negative eight minus four is negative negative eight plus seven 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 seven that's to the left less than or equal to negative seven or x can be equal to or greater than four now to represent this using interval notation it's going to be from negative infinity to negative seven and then union we're going to start back up at four to infinity and we need to use brackets at seven i mean negative seven and four because it include those two points we have a closed circle there so that's how you could 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 zero but we can't have a zero in the bottom of a fraction so this time we can only set the inside just greater than zero so x has to be greater than negative three so the domain is simply going to 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 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 zero so x is equal to and greater than four now we know that in a denominator we can't have a zero so we're going to set it equal or not equal to zero now we could factor it so this is going to be x plus five times x minus five using the difference of squares method so x cannot equal negative five and it can't equal five so now let's make a number line so we have negative five four and five so we're gonna have an open circle at negative five and five and then x is equal to or greater than four so we're gonna have a closer grab four 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 including 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 four using brackets and stop at five using parentheses since it does not include five and then union for the second part is going to go from five 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 a 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 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 has to be only greater than zero but not equal to it because if it's on the bottom it can't be zero so if you have a square root on the top you set it equal to and greater than zero if it's on the bottom simply just greater than zero 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 this because we don't have the underlying symbol it's only 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 like two circles on a number line due to a square root function i like to do a sine 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 multiplied to each other will give us a positive result if we plug in zero zero plus four is positive zero minus four is negative so a 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 squared symbol so it's not going to be anything between negative four and four so for the square root on the bottom x can 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'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 you