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Let's do some example problems dealing with functions and

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their domains and ranges.

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Just as a review, a function is just an operator-- let's

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say this function is f; that tends to be the most typical

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letter for functions-- that operates on some input, in

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this case, the input is x, and it produces some output y.

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Or you could view it as you take some input x, put it into

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your function box f, and it's going to operate on it and

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produce some y.

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And the set of values that x can take on is the domain.

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The set of values that f can legitimately operate on,

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that's the domain.

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And the set of values y that f can

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produce, that is the range.

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Now, with that in mind, let's figure out, one, the function

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definitions for each of these problems here, these example

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problems, and then figure out the domains and the ranges.

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So here we have Dustin charges $10.00 per hour-- so let me

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write that down-- $10.00 per hour for mowing lawns.

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So how much does he charge as a function of hours?

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So let's say Dustin charges, so the function, I'll call it

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d for Dustin.

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Dustin charges as a function of hours.

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It depends on hours.

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The input into our box is going to be

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hours that he worked.

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It's going to be equal to the number of hours times 10.

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It's going to be 10 times-- I won't write the times there.

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Maybe I can just write 10h.

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And, of course, he can't work negative hours, so we could

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write h is going to be greater than or equal to 0.

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That's our function definition.

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If you say, hey, how much is he going to charge for working

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half an hour?

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You put 1/2 in here, which essentially substitutes this h

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with a 1/2.

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You do 10 times 1/2.

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Let me write that down on the side here.

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So Dustin's going to charge for 1/2 hour.

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He's going to charge 10-- wherever you see the h,

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replace it with 1/2-- times 1/2, which is equal to $5.00.

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That's our function definition.

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I was just showing you an example of applying it.

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Now, let's figure out the domain and the range.

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So I almost explicitly say it here, you can't

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work negative hours.

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This number, this function, is not defined if h

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is less than 0.

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So we could say non-negative.

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And we could say real numbers, which it could be everything

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including pi and e and all of that.

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But you can't legitimately charge

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someone 10 times pi dollars.

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That's not a legitimate amount that one can charge someone in

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the real world, because at the end of the day, you have to

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round to the penny.

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You can't actually charge them that.

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So any number that he charges is going to

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be a rational number.

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It can be expressed as a fraction.

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So we can kind of take out numbers like e and pi.

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So if we want to be really cute about it, we would say

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non-negative.

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rational numbers.

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And these are just numbers that can be expressed as a

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fraction, which are most numbers, just not these

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numbers that just keep not repeating and all of that.

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So non-negative rational numbers is what he

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can input in here.

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And then the range, which is the valid-- once again,

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whatever number you put in for h, remember, you can only put

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in non-negative values for h, non-negative rational values

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for h, so whatever numbers you put in for h, you're going to

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get positive values for how much he charges for 10h, for

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the value of the function.

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So once again it's going to be non-negative, and if you put a

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rational number here and it's being expressed as a fraction

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and you multiply it by 10, it's still going to be a

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rational number.

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And if you want to be really cute, you could say any number

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that could be expressed as a dollar sign.

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But anyway, I don't want to get too cute on this problem.

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I think you get the idea.

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Let's do more

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problems. All right.

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Here we have Maria doing some tutoring.

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Maria charges $25.00 per hour for tutoring math with a

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minimum charge of $15.00.

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So Maria charges as a function of hours.

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So she's going to charge $15.00 if you don't get

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enough hours in.

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So at $25.00 per hour, in order to make $15.00, you've

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got to work 15/25, so it's 3/5 of an hour.

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So if her hours are less than 3/5 and greater than or equal

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to 0, she's going to charge $15.00.

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Because if she only worked 1/5 of an hour, the bill would

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have been $5.00, but she says she has a

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minimum charge of $15.00.

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So she's going to charge $15.00 up until 3/5 of an

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hour, or 36 minutes.

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And then after that, for h greater than 3/5, she's going

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to charge $25.00 an hour.

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She's going to charge 25 times h.

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So that's her function definition right there.

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Now, what is the domain, the domain for

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Maria's billing function?

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Well, once again, this is only defined for

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non-negative numbers.

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And by the same logic we used here, we got a little cute.

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We said, oh, she really can't charge e hours or pi hours.

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That's not realistic for her to measure.

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Everything's going to be expressible as a fraction.

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For example, even 5, if you say 5 hours,

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that's still a fraction.

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That's 5 over 1.

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So let's say non-negative rational numbers.

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And what's the range of her charging function or her

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billing function?

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Well, it has to be at least $15.00.

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No matter what, she's going to charge $15.00, so it's going

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to be $15.00 or more.

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So it's going to be rational numbers greater

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than or equal to $15.00.

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There's no situation in which she will charge $14.00.

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There's no situation in which she would

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charge negative $1.00.

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Everything's going to be greater than or equal to 15.

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So now we have these more abstract function definitions,

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so now can stick really any number in it.

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Well, we're not going to deal with

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complexes, any real number.

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But let's see if they limit it a little bit.

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So we have f of x.

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Let me rewrite it. f of x is equal to 15x minus 12.

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So what's the domain here?

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What's our domain?

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Well, I can stick any real number x in here and I'm just

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going to multiply it by 15 and then subtract 12.

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I could put pi there.

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I could put the square of 2 there.

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I could put e there.

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I could do all sorts of crazy things.

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So I'm going to say all real numbers.

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Any real number I put in for x, I'm going to get a

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legitimate output.

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And then what is my range?

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Well, once again, this can take on any value out there.

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I can get to any negative value if I

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make x negative enough.

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I'm going to subtract 12 from it, but I could get to any

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negative value.

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I could get to zero.

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I could get to any positive value.

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So this is also going to be all real numbers.

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Now we have this function.

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It says f of x is equal to-- they forgot the equal sign--

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2x squared plus 5.

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So what is the domain?

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What are all of the valid values for x that I could

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stick here?

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Well, I could stick anything here.

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I could put e there.

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2 times e squared plus 5.

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I could put a square root of 2 there.

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I could put a negative number there.

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So I could really take on any real number.

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All real numbers, positive or negative or otherwise, or even

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non-rational.

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And then what is the range here?

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And this is interesting.

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Because no matter what number I put here, this value right

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here is going to be greater than zero.

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This is going to be, right here, non-negative.

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Even if you put a really negative number here, you're

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going to square it, so it's going to become a really

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positive number when you square it.

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So this expression right here is going to be non-negative.

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So if you take a non-negative number and you add it to 5,

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you're always going to get something greater than or

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equal to 5.

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Worst case, this is a zero, and then you get a 5.

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But if this is either slightly less than zero or slightly

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more than zero, you're going to get values greater than 5.

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So the range is all real numbers greater

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than or equal to 5.

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Your f of x could never take on the value zero.

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There's no way to get zero.

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Problem 5.

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They have f of x is equal to 1 over x.

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So, once again, our domain.

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Well, we could put any value in for x.

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Any real number will work here.

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We're just going to take the inverse of it.

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So this is going to be all reals, all real numbers.

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And then what is the range?

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This is interesting as well.

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Because no matter what I put here, I can put anything here.

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Oh, actually, let me be very, very careful.

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This is not defined for x is equal to 0.

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All real that are not equal to zero.

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If I put zero here, I'm going to get 1/0.

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It will be undefined.

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Let me write that down.

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I almost made a mistake. f of 0 is undefined.

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And that literally means we don't know what

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to do with the zero.

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We have not defined a way to operate on a zero.

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So f of 0 is undefined.

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So the domain is all reals except for zero.

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And now what's the range?

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Well, once again, we can take on any value here except zero.

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We could try to approach zero.

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If x got really, really, really, really, really large,

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then this will approach zero.

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If x got really, really negative, it'll approach zero.

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But in no circumstance will we actually ever get to zero.

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So once again, all reals that are not equal to zero.

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All real numbers except for zero.

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All right.

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Problem 6.

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What is the range of the function y is equal to x

232
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squared minus 5 when the domain is-- so they're

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defining the domain.

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They're restricting it.

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They're saying the domain is just the numbers minus 2,

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minus 1, 0, 1, and 2.

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So this function definition really should say f of x is

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equal to x squared minus 5 for x is equal to-- we could write

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any member for x is a member of the set minus 2,

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minus 1, 0, 1, 2.

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It's not defined for x being anything else to that.

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You don't know what to do with it.

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You can only apply it if x is one of these.

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So what's the range?

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Well, the range is all of the functions, all the values that

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f of x can take on.

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So the range is going to be the set of values that we get

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when we put in all of these different x's.

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So let's try this first one: negative 2.

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Negative 2 squared minus 5.

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That's 4 minus 5.

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That's negative 1.

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Let's do negative 1.

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Negative 1 squared minus 5.

255
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That is 1 minus 5.

256
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That is negative 4.

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You have 0.

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0 squared.

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So f of 0, you have 0 squared minus 5, that's negative 5.

260
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And you have 1.

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00:12:31,559 --> 00:12:34,939
1 squared minus 5, that's negative 4, so it's

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already in our range.

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00:12:36,179 --> 00:12:37,629
Then you have 2.

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00:12:37,629 --> 00:12:41,110
2 squared minus 5, that's negative 1,

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already in our range.

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00:12:42,019 --> 00:12:46,189
So that's all of the values that this function can take

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on, given that we've restricted it

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00:12:47,669 --> 00:12:49,379
to only these inputs.

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