Worked example: domain of algebraic functions

Let's do some more examples finding do mains of functions. So let's say we have a function g of x. So this is our function definition here tells us, look, if we have an input x, the output g of x is going to be equal to 1 over the square root of 6 minus -- we write this little bit neater, 1 over the square root of 6 minus the absolute value of x So like always, pause this video and see if you can figure out what what is the domain of this function. Based on this function definition what is the domain of g? What is the set of all inputs for which this function is defined? Alright. So, to think about all of the inputs that would allow this function to be defined, it may be easier to state when is this function not get defined. Well if we divide by 0 then we're not going to be defined. Or if we have a negative under the radical. So if you think about it -- if if what we have under the radical is 0 or negative -- if it is 0 or negative. If it's 0, you can take the, you can take the principal root of 0. It's going to be 0. But then you are going to divide by 0. That's going to be undefined. And if what you have another radical is negative, the principal root isn't defined for native number, at least the classic principal root is a defined for a negative number. So if 6 minus the absolute value of x is zero or negative, this thing is not going to be defined. Or another way to think about it is it's going to be defined -- so g is is defined -- is defined if -- g is defined if 6 minus, maybe I can write if and only if. Sometimes people write if and only if with two f's right there, iff. g is defined if and only if -- this is kind of mathy way of saying if and only if -- 6 minus the absolute value of x is greater than 0 it has to be, it has to be positive. If it's zero, we're going take the square root of 0 is 0. Then you divide by 0. That's undefined. And if it's less than zero, then you're taking, you're trying to find the principal root of a negative number, that's not defined. So let's see, it's g is defined if and only if this is true. And let's see. We could add the absolute value of x to both sides. We could add the absolute value of x to both sides, then that would give us 6 is greater than the absolute value of x, or that the absolute value of x is less than 6. Or we could say that, you know, let me write that way. The absolute value of x is less than 6. Another way of saying that is x would have to be less than 6 and greater than negative 6. or x is between negative 6 and 6. These two things -- these two things are equivalent. If the magnitude of x is less than 6 then x is greater than negative 6 and less than positive 6. So if we wanted to write the domain in kinda fancy domain-set notation, we could write the domain of g is going to be x, all the x's that are a member of real numbers such that negative 6 is less than x, which is less than 6, and we're done. Let's do another one. And this was gonna get even a little bit a little bit hairier, just for, just for kicks. Alright. So let's say that I have -- let's say that I have -- h of x is equal to -- and I have a kind of composite definition here. So let's say it's x plus 10 over x plus 10 times x minus 9 times x minus 5 times x minus 5 and it's this if it -- h of x is this if x does not equal 5 and it's equal to -- it's equal to pi if x is equal if x is equal to 5. So once again, pause this video. Think about what is the domain of h, or another way to think about is what made h not defined? So let's think about it. So what would make h not defined? So if some -- if x is anything other than 5, we go this clause. If it's 5, we go this clause. So in this clause up here, what would make this thing undefined? Well the most obvious thing is if we divide by 0. So what's going to cause us to divide by 0? So if -- if x is equal to if -- let me write it here -- so we're going to divide by 0 -- divide by 0. That would happen if x is equal to 9. That would happen if x is equal to negative 10. x is equal to negative 10. Now we have to be careful, would that happen if this was the only definition here, it would happen when x equals to 5. But remember when x is equal 5, we don't look at this part of the compound definition. We look at this part. So it's true that up here you would be dividing by 0 if x is equaling 5 but x equaling 5, you wouldn't even look at -- look there. For input is 5, you use this part of the definition. So you'd divide by 0. Maybe I should write it this way, divide by 0 on, I guess you could say the top -- the top clause or the top part a definition. Part of the definition. If x equals 9, X equals negative 10 or -- and that's it because x equal 5 doesn't apply to this top part. If this clause wasn't here then yes, you would write x equals 5. Now we're almost done, but some of you might say, wait, wait, wait, but look, can't I simplify this? I have x plus 10 in the numerator and x plus 10 in the denominator. Can I just simplify this and that will disappear? And you could except, if you did that, you are now creating a different function definition. Because if you just simplify this, you just said 1 over x minus 9 -- 1 over x minus 9 times x minus 5. This is now a different function. That one actually would be defined at x equals negative 10. But the one that we have started with, this one is not. This is -- you're gonna end up with 0 over 0. You gonna end up with that indeterminate form. So for this function, exactly the way it's written, it's not going to be defined with x is equal to 9 or x is equal to negative 10. So once again if you want a fan -- write in our fancy domain set notation. The domain is going to be x all the x's that are a member of the real such that x does not equal 9 and x does not equal negative 10. Any other real number x -- it's going to work including 5. If x equals 5, h of -- h of 5 is going to be equal to pi, because you default to this one over here. h of 5 so OK x is equal 5, we do that one right over there. Now if you gave x equals 9 you gonna divide by 0. If x equals negative 10, you gonna divide by 0, but that's gonna work for anything else. So that right over there is the domain.