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Introduction to functions
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Difference between Equations and Functions
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Function example problems
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Ex: Constructing a function
-
Functions Part 2
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Functions as Graphs
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Understanding function notation
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Positive and negative parts of functions
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Functions (Part III)
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Functions (part 4)
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Sum of Functions
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Difference of Functions
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Product of Functions
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Quotient of Functions
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Evaluating expressions with function notation
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Evaluating composite functions
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Domain of a function
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Domain of a function
Functions Part 2 More examples of solving function problems
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- Welcome to the second presentation on functions.
- So let's take off where we left off before.
- I still apologize -- in retrospect that that
- whole Sal food example.
- Well maybe it was helpful, so I'm going to leave it there.
- Let's do some more problems.
- I think the best thing is to keep doing problems with you
- and I think you'll see the example, and hopefully
- you'll actually see that functions are kind of fun.
- Let's do some more problems.
- Let's start off with an example, not too different
- than what we saw before.
- Let's say that g of x is equal to 1 if x is even, and
- it equals 0 if x is odd.
- And let's say f of x is equal to x plus 3 times g of x.
- And let's say -- I'm going to make it really complicated --
- well, actually I'm not going to make it any more
- complicated now.
- So let's try some problems.
- So let's give an example.
- What is f of 5.
- Well, it's really pretty straightforward.
- We take this 5 and we replace it for x in the function f.
- So f of 5 is equal to 5 plus 3 times g of 5, right?
- We just literally took this 5 and replace it everywhere
- where we see an x.
- If instead of a 5, I had like a dog here, it would be f of dog
- would equal dog plus 3 times g of dog.
- Not that that would necessarily make any sense, but
- you get the idea.
- So f of 5 equals 5 plus 3 times g of 5.
- But what does that equal?
- So the 5 stays the same, plus 3 times -- well what's g of 5?
- Well, if we put 5 here, if 5 is even we do 1,
- if five is odd we do 0.
- Well 5 is odd so it's a 0.
- So g of 5 is equal to 0.
- So this is 3 times 0.
- So this equals just 5, right, because 3 times
- 0 is equal to 5.
- Well what would be f of 6?
- Well, f of 6 would equal 6 plus 3 times g of 6.
- And once again, that equals 6 plus -- well, this time g of
- 6 is, well, 6 is even, so 1.
- So g of 6 is equal to 1.
- So this equals 6 plus 3 times 1.
- So this equals 6 plus 3 which equals 9.
- I think you might be getting the idea now.
- At first when you see a problem with a lot of these functions,
- it seems very confusing.
- But if you just keep taking what's inside of the
- parentheses and replacing that for x and just keep moving
- along that way, you make a lot of progress on these problems.
- Let's try a harder one.
- Let's say I said that f of x is equal to x squared plus 1.
- Let's say that g of x is equal to 2x plus f of x minus 3.
- And h of x is equal to 5x.
- Now I'm going to give you a tough problem.
- What is h of g of x?
- No.
- What is h of g of -- let's pick a number -- let's say 3?
- h of g of 3.
- Actually, we'll do examples in the future where we actually
- could leave the x there and we'll solve for it.
- But let's say this particular example, what is h of g of 3?
- At first you might say wow, this is crazy, Sal, I don't
- know how to even start here.
- But you just take it step-by-step.
- What can we figure out?
- Can we figure out what g of 3 is?
- Well sure.
- We could take the 3 and put it into the function g and
- see what it spits out.
- So let's work on g of 3 first.
- So, g of 3 equals -- well it's 2 times 3, right, we're just
- replacing wherever we see an x with a 3.
- So it's 2 times 3, so that's 6, plus f of -- what, we'll
- just replace the x again.
- 3 minus 3, right?
- So this g of 3 is equal to 6 plus f of what?
- 3 minus 3 is 0.
- Now we have to figure out f of 0 is.
- We have a definition here for f, so we just figure it out.
- f of 0 is equal to -- well, you replace the 0 here.
- So you get 0 squared, which is 0 plus 1.
- So it's f of 0 is 1.
- So you take that and you replace it for f of 0.
- So you get g of 3 is equal to 6 plus 1.
- So g of 3 is equal to 7, right?
- Now we know what g of 3 is equal to.
- We can substitute that back here.
- So that's the same thing -- we know g of 3 is equal to 7,
- so that's the same thing as h of 7.
- And h of 7 is just equal to 5 times 7 equals 35.
- So I think you're probably a little confused here, and I
- would have been if I was in your shoes.
- But the important thing is when you first see this problem
- you're like what can I tackle first?
- h of g of 3, it seems very confusing.
- Well, g of 3, can I tackle that?
- Sure.
- I have a definition of what the function g does when
- it's given an x, or in this case, was given a 3.
- And that's what we did.
- We figured out what g of 3 was first.
- And g of 3, we just substituted the 3, and we said well that's
- 6 plus f of 3 minus 3, right?
- Because we just replaced that x with that 3.
- And we just kept solving.
- We figured out what f of 0 is up here.
- And we got g of 3 equals 7.
- Then we substituted that back in right here.
- We got h of 7 is equal to 35 because it was 5 times 7.
- Let's do some more problems.
- Actually, let's do another example with the same
- set of functions.
- I don't want to keep confusing you with new functions.
- Let me it erase this as fast as I can.
- I think I'm getting faster at this erasing business.
- You can sit and think a little bit about what we just
- did while I erase.
- So let's do another problem.
- What is f of h of 10?
- Well, first we want to figure out what h of 10 is, right?
- Well, we could do it in a different way
- as we'll see later.
- But we can figure out what h of 10 is pretty easily.
- We take the 10, substitute it in for x.
- h of 10 is equal to 5 times x.
- In this case x is 10 so it equals 50.
- So we know h of 10 equals 50.
- So we know h of 10 equals 50, so we substitute
- that back in here.
- So we say f of h of 10 is the same thing as f of 50.
- And then f of 50 is, I think pretty straightforward
- at this point.
- You just take that 50 and replace it back here.
- Well, it's 50 squared plus 1.
- Well, 50 squared is 2,500 plus 1.
- That equals 2,501.
- What is g of h of 1?
- Well, we take h of 1, h of 1 is 5, so this is equal to g of 5.
- And g of 5, we just replace the 5 here, so g of 5 is equal to 2
- times 5 plus f of 5 minus 3.
- We just take wherever we saw an x and replace it with a 5.
- Well, that's equal to 2 times 5 is 10, plus f of 5 minus 3.
- Well 5 minus 3 is 2.
- Plus f of 2.
- What's f of 2?
- Well, 2 squared plus 1 is 5, right? f of 2 is 5
- -- 2 squared plus 1.
- So that equals 10 plus 5 which equals 15.
- If you're still confused, don't worry.
- I'm about to record some more problems that will give you
- even more examples of function problems.
- See you in the next presentation.
- Bye.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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