Tip:
Highlight text to annotate it
X
[ Silence ]
>> This is part 5 of functions, and we're just working
with function notation.
Continuing on.
So we're going to take 2 functions, P of X equals 3X - 2.
And Q of X equals X squared plus 1, and find some things.
Let's do...
Q of X - P of X. How would we do this?
Well Q of X is...
X squared plus 1 -, I need to put this
in parentheses, P of X is 3X - 2.
And now we do our regular old algebra.
X squared plus 1.
Now you have to remember
to distribute my - sign, - 3X plus 2.
So that gives me X squared - 3X.
[ Silence ]
... and then plus 3.
And so there we have Q of X - P of X. Alright what if we wanted
to find P of X times Q of X. Same thing, you need to write P
of X, which is 3X - 2 times Q of X, which is X squared minus 1.
And now you have a binomial times a binomial.
You can use the FOIL method.
So 3X times X squared is 3X cubed.
See the outer term is plus 3X.
The inner term is - 2X squared, and the last terms applied is -
2, and you can leave it like that or I like to write things
in descending order with the exponents.
[ Silence ]
You get this answer.
Alright, what if you're asked to find Q of 2X - 7?
So we have to look at the formula, the rule,
for Q of X. It says take what's in parentheses,
which is your X, square it and add 1.
So I'm going to replace my X with a QX minus 7.
Right? So we'll have to write that QX - 7 in parentheses,
square it, and then add 1.
We don't want to leave an answer like this.
You're going to have to remember how to square a binomial.
Alright, so you could do this a couple of different ways.
You can write it as 2X - 7 times 2X - 7 and use the FOIL method.
You can remember the formula.
But in any case, when you multiply 2X - 7 times 2X -
7 you should get 4X squared - 28X, because if you do the outer
and inner term, if you multiplied it
out the long way you'd have - 14X -
14X plus 49, that's the 7 squared.
I also have this plus 1.
So my final answer will be 4X squared - 28X...
plus 50.
[ Silence ]
Alright why don't you try this one?
P of X plus 8.
Remember, we did this on the previous video.
We're going to plug in X plus H for X. Try this on your own.
So what do we have?
Instead of X we're going to write...
X plus H - 2.
There's 3, whatever is in the parentheses, - 2.
So we have 3X plus 3H - 2.
And that's as far as we could go.
Now what if you are asked to do P
of X plus H - P of X. Try this one.
I'll give you a hint.
You've already figured out P of X plus H, and you're given P
of H so you want to plug those in and simplify it.
Try it on your own.
OK, P of X plus H is what I have right here in the box.
That's 3X plus 3H - 2.
That says subtract, P of X. I look up here...
there's P of X right?
I need to put that in parentheses...
to make sure I subtract the whole quantity P of X.
So this gives me 3X plus 3H - 2.
And what's going to happen over here?
Minus 3X and a plus 2.
Now when you add all the like terms here,
combine the like terms, you end up with simply...
a 3H. Alright, so here's a tough one.
Q of X plus H - Q of X. Alright so,
we're looking at Q of X here right?
That's the..
formula, the rule.
So you're going to have to figure out what Q
of X plus H is, and then subtract that Q of X.
So try this on your own.
This take a little bit of time and space.
Alright so let's do it.
Q of X plus H. Well that means I have to plug
in X plus H for X in the formula.
So that's going to give you an X plus H squared - 1.
Right there that's my Q of X plus H. In fact
to make it simpler I'm going to put this in a different color.
Oops, I'm sorry.
I'm going to put this one in a different color maybe.
I'm having trouble doing either of them.
So I guess I'll leave it.
Minus... technical difficulties, Q of X is the X squared plus 1.
OK? So now I have to simplify this.
X plus H squared.
So you need to multiply X plus H times X plus H. You can do
that on scratch paper and do the FOIL method.
You'll get X squared plus 2XH plus H squared.
You also might know the formula for squaring a binomial,
and that is the formula actually.
Minus 1, minus 1, and now I'm going to distribute this -
sign, 7 - X squared - 1.
Alright now let's see what happens.
So I've got an X squared and I've got a - X squared.
2XH, there's no other term to combine with that.
I have a plus H squared, there's no other term
to combine with that.
And I have a - 1.
Oops, I just noticed a mistake.
Q of X was X squared plus 1,
so this would have been X plus H squared plus 1.
Sorry, so that's a plus 1.
So the 1 and the - 1 did end up cancelling.
Of course if I didn't catch
that mistake I would have had the wrong answer here.
OK, so that's a toughy but this is going to be important formula
for you to be able to work with...
in later classes.
Alright here's another challenging one for you to try.
See if you could complete this.
We're going to do P of -X - Q of -X.
Try it on your own first.
OK P of -X means I'm going to plug in a -X in place of that X.
So I have 3 times -X - 2.
That's your P of -X right here, OK?
Minus, now what's my Q of -X?
I'm going to put a parentheses around it.
I'm going to plug in -X for it's X. So I have to put in a -X
in for the X and square it, and add 1,
and then subtract the whole thing.
So this big thing in parentheses is your Q of -X.
Alright, now...
let's simplify this.
I have 3 times -X, which is -3X, - 2.
Now before distributing my -
sign I need to simplify inside the parentheses.
What's a -X quantity squared?
A -X times a -X is just the same thing as an X squared.
So I'm going to subtract that quantity...
and now I need to distribute my - sign, - X squared - 1.
Almost done.
That'll be...
let's see I'm going to put it in descending order.
-X squared, I have a - 3X, and the -2 and the -
1 is -3 and so that...
would be your final answer.
One more. Alright this should be quick and easy for you.
Try it. What is P of 8 plus H?
OK, what are we going to do?
We're going to plug in 8 plus H for X.
So I have 3 times 8 plus H - 2.
3A plus 3H - 2.
OK, so we've just done lots of problems where you're working
with function notation using the F of X or the G of X,
or the Q of X or the P of X; whatever it is.
And on the previous videos we did it more with numbers.
In this video we did a lot with variables,
more interesting problems.
[ Silence ]