This video is provided as supplementary

material for courses taught at Howard Community College, and in this video I’m going to talk about the composition

of functions. So the composition of functions works

something like this— Let’s say I’ve got two functions, f and g. And function f is

f(x)=3x + 4, and g(x) equals 2x – 5. And then I have a notation

that looks like this… I’ve got parentheses, and inside the parentheses I have

an ‘f’, for function f, and that’s followed by a small open

circle, and then I’ve got a ‘g’, for function g, and that’s the end

of the parentheses. of the parentheses.

And then I may have a parentheses after that that contains a number, like a 3, or maybe it just contains a

variable, like an x. This is a composition of functions. The notation is read as “f compose g of 3”, and the easiest way to understand

composition of functions is going to be to convert this notation into a notation

that looks like this f of g of 3. Now be careful with parentheses. If I have function f, and it’s f of something,

then it’s going to be followed by parentheses. And it was f of g, so that g is going to go inside

the parentheses. But it was g of 3, so the 3 has to go to

into parentheses after the g. Now looking at this what we’ve actually

got is a function of a function. Function f of function g of 3. If we understand that,

it gets fairly easy to solve. Working from the innermost parentheses,

I’m gonna take that g(3) and figure out what it is. I know that g(x) is 2x – 5, so g(3) would mean that I’m going to take that 3

and put it wherever I have an x in g(x). So g(3) is going to be… let’s see

g(3) is going to be 2 times 3, since I originally had 2 times x… minus 5. And 2 times 3 is 6, minus 5… 6 minus 5 is 1. So g(3) is 1. Now I can rewrite that f(g(3)) by replacing the g(3) with the 1. So it’s going to be just f(1) Well, i know what function f is…

Function f is f(x)=3x + 4, so if I want f(1), I’m just going to

put in a 1 wherever the x is. So that’s going to mean it’s 3 times 1 plus 4. 3 times 1 is 3, plus 4 is 7. So f(1) is 7, which means f(g(3) is 7, which means f compose g of 3 is 7. Now I’m going to do this a second time, but I”m going to reverse the

order of my functions. I’m going to do g compose f of 3. Let’s see what happens. ‘g compose f of 3’ could

be rewritten as g of f of 3. Starting from the inner parentheses,

I want to find out what f(3) is. So let’s see… f(x) is 3x + 4, so f(3) is going to be 3 times 3 plus 4. 3 times 3 is 9… plus 4. 9 +4 is 13. So f(3) is 13. That means I can rewrite my composition as g of 13. Now all I have to do is find

out what g(13) is. Well, g(x) is 2x – 5, so g(13) would be 2 times 13 minus 5. 2 times 13 is 26, so that’s 26 minus 5,

and 26 minus 5 is 21. So g(13) is 21, which means that g(f(3)) is 21, and g compose f of 3 is 21. Now I’ve done the same process, and I’ve used the same functions, but I’ve reversed the order of the functions. When you reverse the order of the

functions, you may get two totally different answers.There are

some situations under which you won’t, but want you can’t be sure of that until you do

some checks and some tests, so don’t assume that reversing the order is going to give you the

same answer, there’s a very good chance it won’t. So the order of these two functions is

going to be important. And that’s basically what the composition

of functions is about. Take care, I’ll see you next time.