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Welcome to my presentation on domain of a function.

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So what's is the domain?

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The domain of a function, you'll often hear it combined

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with domain and range.

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But the domain of a function is just what values can I put into

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a function and get a valid output.

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So let's start with something examples.

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Let's say I had f of x is equal to, let's say, x squared.

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So let me ask you a question.

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What values of x can I put in here so I get a valid

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answer for x squared?

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Well, I can really put anything in here, any real number.

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So here I'll say that the domain is the set of x's

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such that x is a member of the real numbers.

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So this is just a fancy way of saying that OK, this r with

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this kind of double backbone here, that just means real

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numbers, and I think you're familiar with real numbers now.

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That's pretty much every number outside of the complex numbers.

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And if you don't know what complex numbers

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are, that's fine.

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You probably won't need to know it right now.

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The real numbers are every number that most people are

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familiar with, including irrational numbers, including

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transcendental numbers, including fractions -- every

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number is a real number.

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So the domain here is x -- x just has to be a member

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of the real numbers.

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And this little backwards looking e or something, this

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just means x is a member of the real numbers.

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So let's do another one in a slight variation.

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So let's say I had f of x is equal to 1 over x squared.

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So is this same thing now?

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Can I still put any x value in here and get

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a reasonable answer?

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Well what's f of 0?

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f of zero is equal to 1 over 0.

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And what's 1 over 0?

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I don't know what it is, so this is undefined.

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No one ever took the trouble to define what 1 over 0 should be.

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And they probably didn't do, so some people probably thought

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about what should be, but they probably couldn't find out with

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a good definition for 1 over 0 that's consistent with

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the rest of mathematics.

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So 1 over 0 stays undefined.

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So f of 0 is undefined.

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So we can't put 0 in and get a valid answer for f of 0.

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So here we say the domain is equal to -- do little brackets,

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that shows kind of the set of what x's apply.

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That's those little curly brackets, I didn't

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draw it that well.

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x is a member of the real numbers still, such that

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x does not equal 0.

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So here I just made a slight variation on what I had before.

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Before we said when f of x is equal to x squared that x

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is just any real number.

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Now we're saying that x is any real number except for 0.

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This is just a fancy way of saying it, and then these curly

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brackets just mean a set.

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Let's do a couple more ones.

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Let's say f of x is equal to the square root of x minus 3.

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So up here we said, well this function isn't defined when we

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get a 0 in the denominator.

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But what's interesting about this function?

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Can we take a square root of a negative number?

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Well until we learn about imaginary and complex

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numbers we can't.

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So here we say well, any x is valid here except for the x's

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that make this expression under the radical sign negative.

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So we have to say that x minus 3 has to be greater than or

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equal to 0, right, because you could have the square to 0,

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that's fine, it's just 0.

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So x minus 3 has to be greater than or equal to 0, so x has to

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be greater than or equal to 3.

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So here our domain is x is a member of the real numbers,

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such that x is greater than or equal to 3.

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Let's do a slightly more difficult one.

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What if I said f of x is equal to the square root of the

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absolute value of x minus 3.

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So now it's getting a little bit more complicated.

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Well just like this time around, this expression of

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the radical still has to be greater than or equal to 0.

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So you can just say that the absolute value of x minus 3 is

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greater than or equal to 0.

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So we have the absolute value of x has to be greater

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than or equal to 3.

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And if order for the absolute value of something to be

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greater than or equal to something, then that means that

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x has to be less than or equal to negative 3, or x has to be

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greater than or equal to 3.

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It makes sense because x can't be negative 2, right?

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Because negative 2 has an absolute value less than 3.

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So x has to be less than negative 3.

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It has to be further in the negative direction than

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negative 3, or it has to be further in the positive

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direction than positive 3.

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So, once again, x has to be less than negative 3 or x

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has to be greater than 3, so we have our domain.

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So we have it as x is a member of the reals

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-- I always forget.

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Is that the line?

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I forget, it's either a colon or a line.

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I'm rusty, it's been years since I've done

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this kind of stuff.

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But anyway, I think you get the point.

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It could be any real number here, as long as x is less

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than negative 3, less than or equal to negative 3, or x is

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greater than or equal to 3.

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Let me ask a question now.

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What if instead of this it was -- that was the denominator,

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this is all a separate problem up here.

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So now we have 1 over the square root of the absolute

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value of x minus 3.

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So now how does this change the situation?

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So not only does this expression in the denominator,

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not only does this have to be greater than or equal to

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0, can it be 0 anymore?

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Well no, because then you would get the square root of 0, which

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is 0 and you would get a 0 in the denominator.

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So it's kind of like this problem plus this

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problem combined.

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So now when you have 1 over the square root of the absolute

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value of x minus 3, now it's no longer greater than or equal to

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0, it's just a greater than 0, right?

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it's just greater than 0.

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Because we can't have a 0 here in the denominator.

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So if it's greater than 0, then we just say greater than 3.

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And essentially just get rid of the equal signs right here.

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Let me erase it properly.

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It's a slightly different color, but maybe

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you won't notice.

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So there you go.

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Actually, we should do another example since we have time.

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Let me erase this.

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OK.

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Now let's say that f of x is equal to 2, if x is even,

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and 1 over x minus 2 times x minus 1, if x is odd.

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So what's the domain here?

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What is a valid x I can put in here.

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So immediately we have two clauses.

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If x is even we use this clause, so f of 4 -- well,

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that's just equal to 2 because we used this clause here.

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But this clause applies when x is odd.

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Just like we did in the last example, what are the

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situations where this kind of breaks down?

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Well, when the denominator is 0.

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Well the denominator is 0 when x is equal to 2, or

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x is equal to 1, right?

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But this clause only applies when x is odd.

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So x is equal to 2 won't apply to this clause.

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So only x is equal to 1 would apply to this clause.

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So the domain is x is a member of the reals, such that

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x does not equal 1.

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I think that's all the time I have for now.

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Have fun practicing these domain problems.

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