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• 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.

• 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

• 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

• 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.

• 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.

Welcome to my presentation on domain of a function.

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# Domain of a function

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Mie Fan posted on 2015/06/10
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