Functions (part 3) Even more examples of function exercises. Introduction of a graph as definition of a function.
Functions (part 3)
- Let's get going with more examples of function problems, and
- hopefully as we keep doing this, you're going to get
- the idea of how all this stuff works.
- So let's do another problem.
- I'll use green this time.
- Let me clear everything.
- So I'll show you-- I showed you that 1, you could define a
- function as just kind of a standard algebraic expression,
- you could also do it a kind of if number is odd, this is what
- you do, if a number is this, is what you do.
- You could also define a function visually.
- Let's say-- let me draw a graph, and I'll use the line
- tool so it's a reasonably neat graph-- that's
- the x-axis there.
- That's pretty good.
- And let's draw the f of x-axis, or you might be used to calling
- that the y-axis, but-- OK.
- I almost had it vertical, but let's see.
- Let's draw a few slashes here.
- And a couple here, like this.
- Sorry if you're getting bored while I draw this graph.
- I should really have some type of tool so that the
- graphs just show up.
- Let me draw a-- let's say that-- let me
- draw this function.
- So this is what?
- This is 1, 2, 3, 4, 5, this is negative 5, this is 5, this
- is 5, this is negative 5.
- And this is x-axis, and this is-- we'll call
- this the f of x-axis.
- Now that might not seem obvious to you at first, but all this
- is saying is let's say when x is equal to negative 5, this
- function-- I'm creating a function definition-- let's say
- it equals 2, that's negative 1, that stays the same, that stays
- the same, then it goes to here, and then it goes to here, to
- here, and then-- let's see.
- I hope I'm not boring you.
- And it just keeps moving up.
- Let me see, what would this look like-- this
- would look like this.
- So if I-- you might think I'm doing something very strange
- right now, but just bear with me while I draw this.
- I hope I don't mess up too much.
- And, see, one like that.
- See one like that.
- So we're like, Sal, this is a very strange looking graph.
- And it is.
- But what this is, is this is a function definition.
- This tells you whenever I input an x, at least for the x's that
- we can see on the graph, this graph tell me what
- f of x equals.
- So if x is equal to negative 5, f of x would equal plus 2.
- And we could draw a couple of examples.
- f of 0, well we go to 0 on the x-axis, and we say
- f of 0 is equal to 0.
- f of 1 is equal to-- well, we go to x equal to 1, and we
- just see where the chart is, well, it equals negative 1.
- I think you get the idea.
- This isn't too difficult, but this is a function definition.
- So we've defined this graph right here as f of x.
- So if that graph-- that's the graph of f of x, and let's say
- that we define g of x is equal to f of x-- let's say
- it's equal to f of x squared minus f of x.
- And let's say that h of x is equal to 3 minus x.
- So what if I were to ask you, what is h of g of negative 1?
- So just like we did in the previous problems, first we'll
- say, well, let's try to figure out what g of negative 1 is,
- and then we can substitute that into h of x.
- So g of negative 1 is equal to-- and this is how I do it.
- There's no trick to it.
- Wherever you see the x, you just substitute it with the
- number that you're saying is now the value for x.
- So you say, well, that's equal to f of negative 1 squared
- minus f of negative 1.
- All I did is at g of negative 1, I just substituted
- it wherever I saw an x.
- Well what's f of negative 1?
- Well, when x is equal to negative 1, f of
- x is equal to 1.
- So f of negative 1-- let's write that, f of negative
- 1 is equal to 1.
- So g of negative 1 is equal to-- well, that's just
- 1 squared minus 1, well that equals 0.
- Because f of negative 1 is 1, so it's 1 squared minus
- 1 that equals 1 minus 1.
- So g of negative 1 is 0, so this is the
- same thing as h of 0.
- Because g of negative 1, we just figured out is 0.
- h of 0, we just take that 0 and substitute it here, so it's 3
- minus 0, so that just equals 3.
- And we solved the problem.
- Let's do another example, and I don't want to erase my graph
- since I took four minutes to actually draw it, let me
- erase what we just did here.
- And what you might want to do after you watch it the first
- time-- and this isn't true just of this video, actually of all
- the videos-- but especially the functions, after watching it
- once, you might want to rewatch it and pause it right after I
- give you the problem and try to do it yourself, and then see--
- and if you get stuck, you can play it, or if you get an
- answer, just you can play the video and make sure that
- we did the same way.
- Let's see.
- I'm going to create another definition
- for g of x this time.
- Let's say that g of x-- oh whoops, I was trying to write
- in black-- let's say that g of x is equal to f of x
- squared plus f of x plus 2.
- So now, in this case, what is g of-- let's pick a random
- number-- what is g of minus-- no, let's pick a, let's
- say-- what is g of minus 2?
- After we try and pick a number that we could find
- an actual solution for.
- Well g of minus 2, wherever we see the x, x is not
- going to be minus 2.
- That is equal to f of minus 2 squared plus
- f of minus 2 plus 2.
- All we did is wherever we saw an x, we substituted
- it, minus 2 there.
- And let's simplify that.
- Well, f of minus 2 squared, we know what minus 2 squared is,
- that's the same thing as f of 4, plus f of minus 2 plus 2.
- That's 0.
- Plus f of 0.
- And now we just figure out what f of 4 and f of 0 is.
- Well, f of 4, we go where x equals r, it's right here,
- and when x equals 4, f of 4 is equal to 2.
- So this is equal to 2 plus f of 0.
- And just as a reminder, this is the definition of f.
- We didn't define it in terms of an algebraic expression, we
- defined in terms of an actual visual graph.
- So what's f of 0? f of 0 is 0.
- When x is equal to 0-- f of 0 is 0 so that's 2 plus 0-- so g
- of negative 2 is equal to 2.
- An interesting thing, you might want to make problems like this
- for yourself and keep experimenting with different
- types of functions, and a very interesting thing would
- actually be to graph g of x, and actually that's a
- good idea, I think.
- I think maybe we'll do that in the future modules to kind of
- play with functions and actually to try graph
- the functions and see how they turn out.
- I will-- I don't know if I have enough time-- actually, I'm
- going to wait until the next lecture to do a couple
- more examples.
- I want to do as many examples on the functions as I can with
- you, because I think as you keep watching and watching the
- function problems and seeing more and more variations on
- functions, you'll see both how general of a concept this is,
- and hopefully you'll get an idea of how the functions
- actually work.
- Well, I'll see you in the next lecture.
- Have fun.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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When naming a variable, it is okay to use most letters, but some are reserved, like 'e', which represents the value 2.7831...
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This is great, I finally understand quadratic functions!
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At 2:33, Sal said "single bonds" but meant "covalent bonds."
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