In this video we're going to talk about composition of functions
And just the notation, they'll write F, circle, G of X
This is not a multiplication symbol. That's why they try it like a little dot. You know denotes.
That's why they try to make it a little bit bigger
Likewise, it's not the word fog
I've heard people ask me about fog x and I was never quite sure what they were talking about so
It's read F of G of x, and what this notation means is you're basically plugging g of x inside of f of x
likewise g of f of x
means take f of x and plug it inside of G
So I think composition is something that [confuses] people a little bit, so let's take a couple examples
so f of x let's make that x squared plus x and
Try to color code things here to make them a little bit easier to follow
suppose we have g of x is let's make [it] 4 minus x
all right and A
Couple things here suppose. We want to figure out f of g of x
[so] again what that means is we want f of g of x
and
Well, I can plug in g of x I know what g of x is
all right g of x is the equation [4] minus x
So that's what I'm going to substitute in there
okay, so there's [f] of g of x and you know kind of off to the side if
somebody said figure out f of 3 what would you do well you would look back at your original f of x equation and
everywhere there was an x
you would replace it with a 3
So I'm going to do the same thing when I do this problem
I'm going to go back and everywhere. There's an x I have to put something in
But this time the thing that I'm substituting in is not just [a] simple number like 3 it
says I have to substitute in the quantity quantity excuse me 4 minus x
so this is the idea with a
composition of functions
Instead of just kind of substituting a simple number in you're putting in a more complicated expression
So let's simplify this one down. This is 4 minus x times 4 minus x
plus just 4 minus x if
I foil this out. I'll get 4 times 4 which is 16
I'll get 4 times negative x which is negative 4x
negative x times 4 that's negative 4x
Negative x times negative x that's positive x squared I have my plus 4 I'm running out of room here
and my minus x still hanging out and
now it's just a matter of combining like terms, and I usually think take things in descending power, so I see an x squared I'll
write him first a
Negative 4x and a negative 4x that's negative 8x
Minus another x that's negative 9x so that takes care [of] my x terms
And then I've got 16 plus 4 which is 20
and that would be the composition f of g of x
Okay, let's maybe take a variation on this one and let's flip it around
So again, I had that f of x is
x squared Plus x
and I had g of x is 4 minus x
suppose now I want to find g of f of x
and
recall multiplication the order in which you do things doesn't matter 2 times 3 is the same as 3 times 2 and
This notation again. It almost looks like
Multiplication, but it's not so you have to be careful because in general you don't get the same thing
So notice I found f of g [of] x to be this stuff
Now let's see what we get when we do g of f of x so
[again] this notation it says you plug f of x into [your] G function
well
again f of x is
The [function] x squared plus 4 excuse me x squared plus x so it says I'm going to plug in x squared
Plus x
and
Just like before if it said find g of 2 I would [find] my G function
everywhere there was an x I would plug in a 2
Well now it says everywhere. There's an x I
Have to plug in all of this mess x squared plus x and this is now going to be the composition function
g of f of x
so if I distribute I'll get negative x squared [I'll]
get a minus x term and then I still have my positive 4 hanging out and that'll be the composition, so
Notice I got something much different
This is what I got when I did f of g of x and
this is what I got when I did g of f of x
so two totally different things there so just something to be aware of
Suppose the original problem was to find g of f of [four]
Let's make it three just so I have a different number floating around
the first thing I would do just like before [I] would just find g of f of x just like it is and
I would get this stuff
And now it says all you're doing is you're [just] replacing the x's with [3S]
So the [x's] with threes, so that's what I would do here and we get 3 squared minus
3 Plus 4
well, you square the 3 first so I'll get 9 but then it will become negative 9 I've got a minus 3 hanging out a
positive 4 well negative 9 [minus] 3 that's negative 12 plus
4 that'll turn out to be negative 8 whoops is all cut off negative 8 and that'll be my solution
So this is the basic idea with composition of functions
You're basically just putting one function inside of the other one
So I hope these couple examples make some sense
I definitely encourage [you] to try to rework them if they're don't if they don't make a lot of sense
Certainly feel free to send me an email. You can visit my website
Just math tutoring [comm] there's lots of other [examples] on there if you need to see some more
Composition of function examples send me an email
let me know what your problems are and I'll try to get to them as soon as I can and
I can post another video with a few more a few [more] examples on there, so
Good luck with this stuff
I think composition is probably of all the kind of operations with functions it can be the most confusing so
Just try a few examples, and I think you'll get there