Determining the domain of advanced functions (Algebra 2 level)
Domain of advanced functions
Welcome to my presentation on domain of a function. So what's is the domain? The domain of a function, you'll often hear it combined with domain and range. But the domain of a function is just what values can I put into a function and get a valid output. So let's start with something examples. Let's say I had f of x is equal to, let's say, x squared. So let me ask you a question. What values of x can I put in here so I get a valid answer for x squared? Well, I can really put anything in here, any real number. So here I'll say that the domain is the set of x's such that x is a member of the real numbers. So this is just a fancy way of saying that OK, this r with this kind of double backbone here, that just means real numbers, and I think you're familiar with real numbers now. That's pretty much every number outside of the complex numbers. And if you don't know what complex numbers are, that's fine. You probably won't need to know it right now. The real numbers are every number that most people are familiar with, including irrational numbers, including transcendental numbers, including fractions -- every number is a real number. So the domain here is x -- x just has to be a member of the real numbers. And this little backwards looking e or something, this just means x is a member of the real numbers. So let's do another one in a slight variation. So let's say I had f of x is equal to 1 over x squared. So is this same thing now? Can I still put any x value in here and get a reasonable answer? Well what's f of 0? f of zero is equal to 1 over 0. And what's 1 over 0? I don't know what it is, so this is undefined. No one ever took the trouble to define what 1 over 0 should be. And they probably didn't do, so some people probably thought about what should be, but they probably couldn't find out with a good definition for 1 over 0 that's consistent with the rest of mathematics. So 1 over 0 stays undefined. So f of 0 is undefined. So we can't put 0 in and get a valid answer for f of 0. So here we say the domain is equal to -- do little brackets, that shows kind of the set of what x's apply. That's those little curly brackets, I didn't draw it that well. x is a member of the real numbers still, such that x does not equal 0. So here I just made a slight variation on what I had before. Before we said when f of x is equal to x squared that x is just any real number. Now we're saying that x is any real number except for 0. This is just a fancy way of saying it, and then these curly brackets just mean a set. Let's do a couple more ones. Let's say f of x is equal to the square root of x minus 3. So up here we said, well this function isn't defined when we get a 0 in the denominator. But what's interesting about this function? Can we take a square root of a negative number? Well until we learn about imaginary and complex numbers we can't. So here we say well, any x is valid here except for the x's that make this expression under the radical sign negative. So we have to say that x minus 3 has to be greater than or equal to 0, right, because you could have the square to 0, that's fine, it's just 0. So x minus 3 has to be greater than or equal to 0, so x has to be greater than or equal to 3. So here our domain is x is a member of the real numbers, such that x is greater than or equal to 3. Let's do a slightly more difficult one. What if I said f of x is equal to the square root of the absolute value of x minus 3. So now it's getting a little bit more complicated. Well just like this time around, this expression of the radical still has to be greater than or equal to 0. So you can just say that the absolute value of x minus 3 is greater than or equal to 0. So we have the absolute value of x has to be greater than or equal to 3. And if order for the absolute value of something to be greater than or equal to something, then that means that x has to be less than or equal to negative 3, or x has to be greater than or equal to 3. It makes sense because x can't be negative 2, right? Because negative 2 has an absolute value less than 3. So x has to be less than negative 3. It has to be further in the negative direction than negative 3, or it has to be further in the positive direction than positive 3. So, once again, x has to be less than negative 3 or x has to be greater than 3, so we have our domain. So we have it as x is a member of the reals -- I always forget. Is that the line? I forget, it's either a colon or a line. I'm rusty, it's been years since I've done this kind of stuff. But anyway, I think you get the point. It could be any real number here, as long as x is less than negative 3, less than or equal to negative 3, or x is greater than or equal to 3. Let me ask a question now. What if instead of this it was -- that was the denominator, this is all a separate problem up here. So now we have 1 over the square root of the absolute value of x minus 3. So now how does this change the situation? So not only does this expression in the denominator, not only does this have to be greater than or equal to 0, can it be 0 anymore? Well no, because then you would get the square root of 0, which is 0 and you would get a 0 in the denominator. So it's kind of like this problem plus this problem combined. So now when you have 1 over the square root of the absolute value of x minus 3, now it's no longer greater than or equal to 0, it's just a greater than 0, right? it's just greater than 0. Because we can't have a 0 here in the denominator. So if it's greater than 0, then we just say greater than 3. And essentially just get rid of the equal signs right here. Let me erase it properly. It's a slightly different color, but maybe you won't notice. So there you go. Actually, we should do another example since we have time. Let me erase this. OK. Now let's say that f of x is equal to 2, if x is even, and 1 over x minus 2 times x minus 1, if x is odd. So what's the domain here? What is a valid x I can put in here. So immediately we have two clauses. If x is even we use this clause, so f of 4 -- well, that's just equal to 2 because we used this clause here. But this clause applies when x is odd. Just like we did in the last example, what are the situations where this kind of breaks down? Well, when the denominator is 0. Well the denominator is 0 when x is equal to 2, or x is equal to 1, right? But this clause only applies when x is odd. So x is equal to 2 won't apply to this clause. So only x is equal to 1 would apply to this clause. So the domain is x is a member of the reals, such that x does not equal 1. I think that's all the time I have for now. Have fun practicing these domain problems.