Find a function

Question

If g(x) = 2x + 1 and h(x) = 4x² + 4x + 8, find a function f such that f ∘ g = h.

2.If f(x) = 5x + 6  and h(x) = 5x² + 5x + 1, find a function g such that f ∘ g = h.

Answer 1

For part 1:

f(2x+1) = 4x²+4x+8; since (2x+1)²= 4x²+4x+1, looks like we just need to add on a +7.

f(x) = x²+7

f(2x+1) = (2x+1)²+7= 4x²+4x+1+7=4x²+4x+8

For part 2:

f(x) = 5x+6; we want f(/) = 5x²+5x+1, ;let's try putting just x²+x into f(x).

f(x²+x) = 5(x²+x) + 6 = 5x²+5x+6 -- too big by 5, so let's try x²+x-1.

g(x) = x²+x-1

f(g(x)) = 5(x²+x-1) + 6 = 5x²+5x-5+6 = 5x²+5x+1