Composition and Decomposition of Functions PreCalculus Composition and Decomposition of Functions

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

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

Example 3 4 3 1 –2.5 –2 2 7) f(–3) 8) f(0) 9) g(8) 10) f(g(–2)) 11) g(f(4)) 12) (f°g) (8)

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

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) = ___

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))

Example 5 Decompose: