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This video is going to be an introduction to function notation.
So let's see what that's all about.
Here I've got two equations. You're used to seeing things like this.
I've got y = 3x + 5
and y = x-squared - 2x.
Now
if I want to deal with both of these equations at the same time,
and I want to talk about them, I have to either let you know which one I mean
by reading the whole equation
or by saying "the first equation" and "the second equation".
And if I had more equations, like another y equals something and yet
another one, I'd have to talk about the third equation and the fourth equation.
So this gets kind of clumsy at times.
What we're going to do
is we're going to give
names to each of the equations,
and the names are going to be letters. So the first equation, y =
3x + 5,
I'm going to use an 'f' for that one.
And I'm going to write the 'f', and then in parentheses I'm going to put an x,
and then equals whatever that 'y' used to equal, 3x + 5.
So, if y equals 3x + 5, and f(x), this is read as "f of x",
equals 3x + 5, then y must equal f(x), this first y.
Now I'm going to take the second equation and give that a different name.,
basically, a different letter.
So I can call that 'g'.
So this is going to be read as "g of x"
equals... and then I'll just copy what the old 'y' used to equal, x-squared - 2x.
And now I can talk about function f and function g.
Now I could have any letters I want instead of 'f' and 'g',
just like you can other letters instead of 'x' and 'y'.
Typically,
when we've got no reason to do anything differently, we start out with the letter
'f',
and then if we have another function, we'll call that 'g',
the one after than might be 'h', and so on.
Okay, so
the first use for function notation
is it allows you to tell which equation you're talking about,
which function you're talking about,
either the f-function or the g-function.
Now there's another nice thing about this.
Let's take that first one, f(x)
equals 3x + 5.
If instead of that, I had you the y = 3x + 5 and I and
I wanted to know what happens when x is 2, I would have to say "evaluate y when
x equals two".
Well a faster way to do that with function notation
is to
replace the x
in the f(x)
with whatever I want to x to equal.
So if I write "f of
2",
then what that means is I'm going to find out
what 3x + 5 is when instead of having an x, I have a 2.
So all I'll do is put than 2 in wherever I have an x
on the right side of my function.
So that's going to be 3
times 2
plus 5.
And of course when I multiply that out, I'll have 3 times 2 is 6, and 6 plus 5 is 11.
Now it can get even fancier than that.
Let's say
I take function g. So, function g
was
g(x) equals
x-squared minus
2x. And if I want to know what that function it is
when x equals
let's say 5,
I'll just write g(5),
and I'll take
each of my x's in the right side of the function
and replace it with a 5, whatever I have in parentheses here
after the name of my function.
So that means I'm going to have 5-
squared
minus
2
times 5.
Let's see,
5-squared is 25,
and 2 times 5 is 10,
so g(5)
equals
15.
Let's try one a little more complicated than that.
Let's say
that instead of having
the x,
I want to see what would happen if the x was bigger, what would happen if
every x was bigger by 2. So in other words,
I want to find out what happens
when, instead of x, I've got x + 2.
So I'm going to write
g(x + 2),
and let's see what that equals.
Well, that means instead of the x-squared that I have,
I'm going to take that whole 'x + 2'
and I'll square it,
and then I'm going to subtract 2
times
x + 2.
And now I'll work this out.
So let's see. (x + 2)-squared means that (x + 2) times
(x + 2)
minus
2 times
(x + 2).
So that means I'm going to foil the first two binomials here. So I have x times
x is x-squared,
x times 2 is 2x,
I've got a 2 times x, that's another 2x,
2 times 2 is 4.
And I'm subtracting,
so I'll distribute this 2.
2 times x is
-2x,
and -2 times 2 is -4.
And then when I combine all my like terms,
I'm going to have
x-squared.
I've got a 2x and a 2x and a -2x. Well, -2x will cancel out one of the
2x's, so I'll have x-squared
plus 2x.
And then I've got a +4 and a -4, so they'll cancel each other out,
and I'm going to get x-squared + 2x. So, g(x + 2)
equals
x-squared
plus 2x.
Okay, just reviewing this,
I'm using function notation so I can tell the difference between difference
equations that I have.
I'm naming the equations,
and I'm naming them with letters, like the letter 'f' or the letter 'g',
or whatever letter I want.
So for the first equation,
basically I'm saying that f(x) is the same as 'y'.
And for the second equation, I'm just saying that g(x) is the same as 'y'.
So I just replace one with the other. Then I can use function notation
to find out
what that function would equal
if I replaced the x
with something more specific,
either a number or some
variation on the x, like 'x + 2'.
So that's basically all that's going on here.
It's a little bit hard to get used to it first,
but just do a bunch of exercises and after a while it becomes more or less second nature.
That's it for now.
Take care, I'll see you next time.