If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains ***.kastatic.org** and ***.kasandbox.org** are unblocked.

Main content

Current time:0:00Total duration:6:45

CCSS.Math:

let's have a little bit of a review of what a function is before we talk about what it means it what what the domain of a function means so a function we can view as something so I'll put a function in this box here and it takes inputs and for a given input it's going to produce an output which we call f of X so for example let's say that we have the function let's say we have the function f of X is equal to 2 over X so in this case if so let me see so if that's my function f if I were to input the number 3 well f of 3 that we're going to output we have we know how to figure that out we've defined it right over here it's going to be equal to 2 over 3 it's going to be equal to 2 over 3 so we were able to for that input we're able to find an output if our input was PI then we input into our function and then F of Pi when when X is PI we're going to output f of Pi which is equal to 2 over PI so we could write this as 2 over 5 so we were able to find the output pretty easily but now let's do something interesting let's attempt to input 0 into the function if we input 0 does the function tell us what we need to output does this definition tell us what we need to output so if I attempt to put x equals 0 then this definition would say f of 0 would be 2 over 0 but 2 over 0 is undefined let me write this 2 over 0 this is undefined this function definition does not tell us what to actually do with 0 it gives us an undefined answer so this function is not defined here it gives a question mark so this gets to the essence of what domain is domain is the set of all inputs over which the function is defined so the domain of this function f would be all real numbers except for x equals 0 so let me write down these big ideas this is the domain a domain of a function actually my write that out domain of a function a domain of a function is set of all inputs inputs over which the function is defined over which the function is defined or the function has defined outputs over which the function has defined outputs so the domain for this F in particular so the domain for this one if I wanted to say its domain I could say look it's going to be the set of this curly brackets these are kind of typical mathy set notation I can say okay is going to be the set of and I'm going to put curly brackets like that well X can be a member this is this little this little symbol means a member of the real numbers but it can't just be any real number can be most of the real numbers except it cannot be zero because we don't know this definition it's undefined when you put it in put a zero so X is a member of the real numbers and we write real numbers we write it with this kind of double stroke right over here that's the set of all real numbers such that but we have to we have to put it up put the exception zero is not a x equals zero is not a member of that domain such that X does not does not equal zero now let's let's make this a little bit more concrete by doing some more examples so more examples we do hopefully the clearer this will become so let's say we have another function and just be clear when at all we don't always have to use F F sin X's we could say let's say we have G of Y is equal to the square root of Y minus six so what's the domain here what is the set of all inputs over which this function G is defined so here we are inputting a Y into function G and we're going to output G of Y well it's going to be defined as long as whatever we have under the radical right over here is non-negative if this becomes negative are our traditional principal root operator here is not defined we need something that if we if this ended up being a negative number all right well how you take the principal root of a negative number and we're just saying this is kind of the traditional principal root up Reiter so y minus six Y minus six needs to be greater than or equal to zero in order for in order for G to be defined for that input Y or you could say add six to both sides Y needs to be greater than or equal to six or you can say G is defined for any inputs Y that are greater than or equal to six so we could say the domain here we could say the domain here is the set of all Y's that are a member of the real numbers such that Y such that they're also greater than or equal such that they're also greater than or equal to six so hopefully this is starting to make some sense and you know we're also always used to functions defined this way but you could even see functions that are defined in fairly exotic ways you could see a function let me say H of X H of X could be defined as it literally could be defined as well H of X is going to be one if X is equal to PI and it's equal to 0 if if X is equal to 3 now what's the domain here and I encourage you to pause the video and think about it well this function is actually only defined for two inputs if you we know H of we know H of Pi if you input PI into it we know you're going to output 1 and we know that if you input 3 into it H of 3 when x equals 3 you're going to you're going to put some commas here you're going to get a 0 but if you input anything else what's H of 4 going to be well it hasn't defined it it's undefined what's H of negative 1 going to be it hasn't defined it so the domain the domain here the domain of H is literally its it's just literally going to be the the two valid inputs that X can be are 3 and PI 3 and by these are the only valid inputs these are the only two numbers over which this function is actually defined so this hopefully starts to give you a flavor of of why we care about domains not all functions are defined over all real numbers some are defined for only a small subset of real numbers or for some other thing or only whole numbers or natural numbers or positive numbers negative numbers or they have exceptions so we'll see that as we do more and more examples