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• A function-- and I'm going to speak about it

• in very abstract terms right now-- is something that

• will take an input, and it'll munch on that input,

• it'll look at that input, it will do something

• to that input.

• And based on what that input is, it will produce a given output.

• What is an example of a function?

• I could have something like f of x-- and x

• tends to be the variable most used for an input

• into the function.

• And the name of a function, f tends

• to be the most-used variable.

• But we'll see that you can use others-- is equal to,

• let's say, x squared, if x is even.

• And let's say it is equal to x plus 5, if x is odd.

• What would happen if we input 2 into this function?

• The way that we would denote inputting 2

• is that we would want to evaluate f of 2.

• This is saying, let's input 2 into our function f.

• And everywhere we see this x here,

• this variable-- you can kind of use as a placeholder-- let's

• replace it with our input.

• So let's see.

• If 2 is even, do 2 squared.

• If 2 is odd, do 2 plus 5.

• Well, 2 is even, so we're going to do 2 squared.

• In this case, f of 2 is going to be 2 squared, or 4.

• Now what would f of 3 be?

• Well, once again, everywhere we see this variable,

• we'll replace it with our input.

• So f of 3, 3 squared if 3 is even, 3 plus 5 if 3 is odd.

• Well, 3 is odd, so it's going to be 3 plus 5.

• It is going to be equal to 8.

• You might say, OK, that's neat, Sal.

• This was kind of an interesting way

• to define a function, a way to kind of munch on these numbers.

• But I could have done this with traditional equations

• in some way, especially if you allowed

• me to use the squirrelly bracket thing.

• What can a function do that maybe my traditional toolkits

• might have not been as expressive about?

• Well, you could even do a function like this.

• Let me not use f and x anymore, just

• to show you that the notation is more general than that.

• I could say h of a is equal to the next largest number that

• starts with the same letter as variable a.

• And we're going to assume that we're dealing in English.

• Given that, what is h of 2 going to be?

• Well, 2 starts with a T. What's the next largest

• number that starts with a T?

• Well, it's going to be equal to 3.

• h of 8 be equal to?

• Well, 8 starts with an E. The next largest number

• that starts with an E-- it's not 9, 10-- it would be 11.

• And so now you see it's a very, very, very general tool.

• This h function that we just defined, we'll look at it.

• We'll look at the letter that the number

• starts with in English.

• So it's doing this really, really, really, really

• wacky thing.

• Now not all functions have to be this wacky.

• In fact, you have already been dealing with functions.

• You have seen things like y is equal to x plus 1.

• This can be viewed as a function.

• We could write this as y is a function

• of x, which is equal to x plus 1.

• If you give as an input-- let me write

• it this way-- for example, when x is 0 we could say f of 0

• is equal to, well, you take 0.

• It's equal to 1.

• f of 2 is equal to 2.

• You've already done this before.

• You've done things where you said, look,

• let me make a table of x and put our y's there.

• When x is 0, y is 1.

• I'm sorry.

• I made a little mistake.

• Where f of 2 is equal to 3.

• And you've done this before with tables where you say,

• look, x and y.

• When x is 0, y is 1.

• When x is 2, y is 3.

• You might say, well, what was the whole point of using

• the function notation here to say

• f of x is equal to x plus 1?

• The whole point is to think in these more general terms.

• For something like this, you didn't really

• have to introduce function notations.

• But it doesn't hurt to introduce function notations because it

• makes it very clear that the function takes an input,

• takes my x-- in this definition it munches on it.

• It says, OK, x plus 1.

• And then it produces 1 more than it.

• So here, whatever the input is, the output is 1 more

• than that original function.

• Now I know what you're asking.

• All right.

• Well, what is not a function then?

• Well, remember, we said a function

• is something that takes an input and produces only one

• possible output for that given input.

• For example-- and let me look at a visual way

• of thinking about a function this time, or a relationship,

• I should say-- let's say that's our y-axis,

• and this right over here is our x-axis.

• Let me draw a circle here that has radius 2.

• So it's a circle of radius 2.

• This is negative 2.

• This is positive 2.

• This is negative 2.

• So my circle, it's centered at the origin.

• That's my best attempt at drawing the circle.

• Let me fill it in.

• So this is a circle.

• The equation of this circle is going

• to be x squared plus y squared is equal to the radius squared,

• is equal to 2 squared, or it's equal to 4.

• The question is, is this relationship between x and y--

• here I've expressed it as an equation.

• Here I've visually drawn all of the x's and y's that satisfy

• this equation-- is this relationship between x

• and y a function?

• And we can see visually that it's not

• going to be a function.

• You pick a given x.

• Let's say x is equal to 1.

• There's two possible y's that are associated with it,

• this y up here and this y down here.

• We could even solve for that by looking at the equation.

• When x is equal to 1, we get 1 squared plus y squared

• is equal to 4.

• 1 plus y squared is equal to 4.

• Or subtracting 1 from both sides, y squared is equal to 3.

• Or y is equal to the positive or the negative square root of 3.

• This right over here is the positive square root of 3,

• and this right over here is the negative square root of 3.

• So this situation, this relationship

• where I inputted a 1 into my little box here,

• and associated with the 1, I associate

• both a positive square root of 3 and a negative square root

• of 3, this is not a function.

• I cannot associate with my input two different outputs.

• I can only have one output for a given input.

A function-- and I'm going to speak about it

A2 初級

# 関数とは何か？| 関数とそのグラフ｜代数II｜カーンアカデミー (What is a function? | Functions and their graphs | Algebra II | Khan Academy)

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piano.man に公開 2021 年 01 月 14 日