Question

Composition of Functions

If f(x) = 2x+1
find

A. f o f(x)
f(f(x)) = f(2x+1)
f(2x+1) = 2(2x+1)
f(f(x)) = 4x +2

B. f o f o f o f(x)

^ I don't get how to solve this one

C. f o f o f o f o f ...o f(x) n functions of f

^ I also don't get how to solve this one

given f(x) = ax+b and g(x) = cx+d and
f o g(x) = g o f(x), what is the relationship between a,b,c,d?

Answer 1

f(x) = 2x+1
f(f) = 2f+1 = 2(2x+1)+1 = 4x+3
f(f(f)) = 2(4x+3)+1 = 8x + 7
f(f(f(f(x)))) = 2(8x+7)+1 = 16x + 15

fⁿ(x) = 2^n * (x+1) - 1

f(g) = ag+b = a(cx+d)
g(f) = cf+d = c(ax+b)
so, if f(g) = g(f)

acx + ad = acx + bc
ad = bc