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- In this video, I want to do a few examples
- dealing with functions.
- Functions tend to be something that a lot of students find
- difficult, but I think if you really get what we're talking
- about, you'll see that it's actually a pretty
- straightforward idea.
- And you sometimes wonder, well what was all
- of the hubbub about?
- All a function is, is an
- association between two variables.
- So if we say that y is equal to a function of x, all that
- means is, you give me an x.
- You can imagine this function is kind of eating up this x.
- You pop an x into this function.
- This function is just a set of rules.
- It's going to say, oh, with that x, I
- associate some value y.
- You can imagine it is some type of a box.
- That is a function.
- When I give it some number x, it'll give me some
- other number y.
- This might seem a little abstract.
- What are these x's and y's?
- Maybe I have a function-- let me make it like this.
- Let's say I have a function definition
- that looks like this.
- For any x you give me, I'm going to produce 1 if x is
- equal to-- I don't know-- 0.
- I'm going to produce 2 if x is equal to 1.
- And I'm going to produce 3 otherwise.
- So now we've defined what's going on inside of the box.
- So let's draw the box around it.
- This is our box.
- This is just an arbitrary function definition, but
- hopefully it'll help you understand what's actually
- going on with a function.
- So now if I make x is equal to-- if I pick x is equal to
- 7, now what is f of x going to be equal to?
- What is f of 7 going to be equal to?
- So I take 7 into the box.
- You could view it as some type of a computer.
- The computer looks at that x and then looks at its rules.
- It says, OK. x is 7.
- Well x isn't 0. x isn't 1.
- I go to the otherwise situation.
- So I'm going to pop out a 3.
- So f of 7 is equal to 3.
- So we write f of 7 is equal to 3.
- Where f is the name of this function, this rule system, or
- this association, this mapping, whatever you
- want to call it.
- When you give it a 7, it'll produce a 3.
- When you give f a 7, it'll produce a 3.
- What is f of 2?
- Well, that means instead of x is equal to 7, I'm going to
- give it an x equal 2.
- Then the little computer inside the function is going
- to say, OK, let's see, when x is equal to 2.
- No, I'm still in the otherwise situation.
- x isn't 0 or 1.
- So once again f of x is equal to 3.
- So, this is f of 2 is also equal to 3.
- Now what happens if x is now equal to 1?
- Well then it's just going to turn over this.
- So f of 1.
- It's going to look at its rules right here.
- Oh look, x is equal to 1.
- I can use my rule right here.
- So when x is equal to 1, I spit out a 2.
- So f of 1 is going to be equal to 2.
- I spit out f of 1, which is equal to 2 in that situation.
- That's all a function is.
- Now, with that in mind, let's do some of these example
- problems. They tell us for each of the following
- functions, evaluate these different functions-- these
- are the different boxes they've created-- at these
- different points.
- Let's do part a first. They're defining the box.
- f of x is equal to negative 2x plus 3.
- They want to know what happens when f is equal to negative 3.
- Well f is equal to negative 3, this is telling me what do I
- do with the x?
- What do I produce?
- Wherever I see an x, I replace it with the negative 3.
- So it's going to be equal to negative 2.
- Let me do it this way, so you see exactly what I'm doing.
- That negative 3, I'll do it in that bold color.
- It's negative 2 times negative 3 plus 3.
- Notice wherever there was an x, I put the negative 3.
- So I know what the black box is going to produce.
- This is going to be equal to negative 2 times negative 3 is
- 6 plus 3, which is equal to 9.
- So f of negative 3 is equal to 9.
- What about f of 7?
- I'll do the same thing one more time. f of-- I'll do 7 in
- yellow-- f of 7 is going to be equal to negative 2
- times 7 plus 3.
- So this is equal to negative 14 plus 3, which is equal to
- negative 11.
- You put in-- let me make it very clear-- you put in a 7
- into our function f here and it will pop out a negative 11.
- That's what this just told us right there.
- This is the rule.
- This is completely analogous to what I did up here.
- This is the rule of our function.
- Let's do the next two.
- I won't do part b.
- You can do part b for fun.
- I'll do part c after that, just for the sake of time.
- Now we are at f of 0.
- Here I'll just do it in one color.
- I think you're getting the idea. f of 0.
- Wherever we see an x, we put a 0.
- So negative 2 times 0 plus 3.
- Well, that's just going to be a 0.
- So f of 0 is 3.
- Then one last one. f of z.
- They want to keep it abstract for us.
- Here I'll color code it.
- So f of z.
- Let me make the z in a different color.
- f of z.
- Everywhere that we saw an x, we will now
- replace it with a z.
- Negative 2.
- Instead of an x, we're going to put a z there.
- We're going to put an orange z there.
- Negative 2 times z plus 3.
- And that's our answer. f of z is negative 2z plus 3.
- If you imagine our box, the function f.
- You put in a z, you are going to get out a negative 2 times
- whatever that z is plus 3.
- That's all this is saying.
- It's a little bit more abstract, but same exact idea.
- Now let's just do part c here.
- Let me clear this actually.
- I'm running out of real estate.
- Let me clear all of this business.
- Let me clear all of this business.
- We can do part c.
- I'm skipping part b.
- You can work on that part.
- Part b.
- They tell us-- this is our function definition.
- Sorry, I said I was doing part c.
- This is our function definition.
- f of x is equal to 5 times 2 minus x over 11.
- So let's apply these different values of x, these different
- inputs into our function.
- So f of negative 3 is equal to 5 times 2 minus-- wherever we
- see an x, we put a negative 3.
- 2 minus negative 3 over 11.
- This is equal to 2 plus 3.
- This is equal to 5.
- So you get 5 times 5 over 11.
- That's equal to 25/11.
- Let's do this one.
- F of 7.
- For this second function right here, f of 7 is equal to 5
- times 2 minus-- now for x we have a 7.
- 2 minus 7 over 11.
- So what is this going to be equal to?
- 2 minus 7 is negative 5.
- 5 times negative 5 is negative 25/11.
- Then finally, well we have two more. f of 0.
- That's equal to 5 times 2 minus 0 So this is just 2.
- 5 times 2 is 10.
- So this is equal to 10/11.
- One more.
- f of z.
- Well every time we saw an x, we're going to
- replace it with a z.
- It's equal to 5 times 2 minus z over 11.
- And that's our answer.
- We could distribute the 5.
- You could say this is the same thing as 10 minus 5z over 11.
- We could even write it in slope-intercept form.
- This is the same thing as minus 5/11 z plus 10/11.
- These are all equivalent.
- But that is what f of z is equal to.
- Now.
- A function, we said, if you give me any x value, I will
- give you an output.
- I will give you an f of x.
- So if this is our function, you give me an x, it will
- produce an f of x.
- It can only produce 1 f of x for any x.
- You can't have a function that could produce two possible
- values for an x.
- So you can't have a function-- this would be an invalid
- function definition-- f of x is equal to 3 if
- x is equal to 0.
- Or it could be equal to 4 if x is equal to 0.
- Because in this situation, we don't know what f of 0 is.
- What it's going to be equal?
- It says if x is equal to 0, it should be 3 or it could be--
- we don't know.
- We don't know.
- We don't know.
- This is not a function even though it might
- have looked like one.
- So you can't have two f of x values for one x value.
- So let's see which of these graphs are functions.
- To figure that out, you could say, look at any x value
- here-- pick any x value-- I have exactly one f of x value.
- This is y is equal to f of x right here.
- I have exactly only one-- at that x, that
- is my y value here.
- So you could have a vertical line test, which says at any
- point if you draw a vertical line-- notice a vertical line
- is for a certain x value.
- That shows that I only have one y value at that point.
- So this is a valid function.
- Any time you draw a vertical line, it will only intersect
- the graph once.
- So this is a valid function.
- Now what about this one right here?
- I could draw a vertical line, let's say, at
- that point right there.
- For that x, this relation seems to have two
- possible f of x's.
- f of x could be that value or f of x could be that value.
- Right?
- We're intersecting the graph twice.
- So this is not a function.
- We're doing exactly what I described here.
- For a certain x, we're describing two possible y's
- that could be equal to f of x.
- So this is not a function.
- Over here, same thing.
- You draw a vertical line right there.
- You're intersecting the graph twice.
- This is not a function.
- You're defining two possible y values for 1 x value.
- Let's go to this function.
- It's kind of a weird looking function.
- Looks like a reversed check mark.
- But any time you draw a vertical line, you're only
- intersecting it once.
- So this is a valid function.
- For every x, you only have one y associated.
- Or only one f of x associated with it.
- Anyway, hopefully you found that useful.