1. Composition and Decomposition of Functions
PreCalculus
Composition and Decomposition of Functions

2. Example 1 1 3
1) f(–2) ) f(1) ) f(7)

3. Example 2 4 –1 2
4) f(–6) ) f(4) ) f(8)

4. Example 3 4 3 1 –2.5 –2 2 7) f(–3) 8) f(0) 9) g(8)
10) f(g(–2)) ) g(f(4)) ) (f°g) (8)

5. Example 4 - Composition of Functions
Evaluate each of the following:
1) D: f(x) = ___ 2) D: g(x) = ___ 3) D: h(x) = ___

6. Evaluate each of the following:
Example 4 - Cont.
Evaluate each of the following:
4)a) f(g(x)) = ___ b) D: f(g(x)) = ___
5)a) f(h(x)) = ___ b) D: f(h(x)) = ___
6)a) (g°h)(x) = ___ b) D: (g°h)(x) = ___

7. Decomposition of Functions
Sometimes a function is a composite function and we will need to break it into its constituent functions. This will be EXTREMELY important in Calculus.
Decompose: f(x) = sin (x2)
To decide how to start: I imagine that I am replacing x with a number. The first operation that I would do to that number is the innermost function, then I continue outward until I have no operations (functions left).
In this case, this first thing I would do when I entered a number for x is to square it. I would then take the sine of the result and there are no other operations. Therefore, the innermost function is x2 and the outer function is sine x.
a(x) = sin (x)
b(x) = x2
Therefore: f(x) = a(b(x))

8. Example 5
Decompose: