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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.

  • 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.

Welcome to my presentation on domain of a function.

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A2 US equal domain greater denominator negative square root

Domain of a function

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    Mie Fan posted on 2015/06/10
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