Answer:
Composition of Functions
"Function Composition" is applying one function to the results of another:
Function Composition
The result of f() is sent through g()
It is written: (g º f)(x)
Which means: g(f(x))
Example: f(x) = 2x+3 and g(x) = x2
"x" is just a placeholder. To avoid confusion let's just call it "input":
f(input) = 2(input)+3
g(input) = (input)2
Let's start:
(g º f)(x) = g(f(x))
First we apply f, then apply g to that result:
Function Composition
(g º f)(x) = (2x+3)2
What if we reverse the order of f and g?
(f º g)(x) = f(g(x))
First we apply g, then apply f to that result:
Function Composition
(f º g)(x) = 2x2+3
We get a different result!
When we reverse the order the result is rarely the same.
So be careful which function comes first.
Symbol
The symbol for composition is a small circle:
(g º f)(x)
It is not a filled in dot: (g · f)(x), as that means multiply.
Composed With Itself
We can even compose a function with itself!
Example: f(x) = 2x+3
(f º f)(x) = f(f(x))
First we apply f, then apply f to that result:
Function Composition
(f º f)(x) = 2(2x+3)+3 = 4x + 9
We should be able to do it without the pretty diagram:
(f º f)(x) = f(f(x))
= f(2x+3)
= 2(2x+3)+3
= 4x + 9
Domains
It has been easy so far, but now we must consider the Domains of the functions.
domain and range graph
The domain is the set of all the values that go into a function.
The function must work for all values we give it, so it is up to us to make sure we get the domain correct!
Example: the domain for √x (the square root of x)
We can't have the square root of a negative number (unless we use imaginary numbers, but we aren't), so we must exclude negative numbers:
The Domain of √x is all non-negative Real Numbers
On the Number Line it looks like:
zero onwards
Using set-builder notation it is written:
{ xmember ofreals | x ≥ 0}
Or using interval notation it is:
[0,+∞)
