Placeholder Image

字幕表 動画を再生する

  • 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.

  • Now what would h of-- I don't know, let's think about this,

  • 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.

  • You add 1.

  • 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.

  • It has radius 2.

  • 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

字幕と単語

動画の操作 ここで「動画」の調整と「字幕」の表示を設定することができます

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)

  • 54 1
    piano.man に公開 2021 年 01 月 14 日
動画の中の単語