Pre Calculus Composition and Decomposition of Functions Example

Pre. Calculus Composition and Decomposition of Functions

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

Example 2 4) f(– 6) 4 5) f(4) – 1 6) f(8) 2

Example 3 7) f(– 3) 10) f(g(– 2)) – 2 4 8) f(0) 3 9) g(8) 1 11) g(f(4)) – 2. 5 12) (f°g) (8) 2

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

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) = ___

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 (x 2) 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 x 2 and the outer function is sine x. a(x) = sin (x) b(x) = x 2 Therefore: f(x) = a(b(x))

Decompose: Example 5