Question

I was assigned this problem and I'm coming up against a brick wall. Here it is: Given that f(x) = 2x+1 and h(x) = 2x^2+4x+1, find a function g such that f(g(x)) = h(x)

I tried to factor 2x^2+4x+1 but ended up with something that didn't work. Am I missing something here? I'm so lost.

Thanks in advance,

Alyn

Answer 1

since the resulting h(x) is quadratic, g(x) must have been quadratic

let g(x) = ax^2 + bx + c

then
f(g(x)) = 2(ax^2 + bx + c) + 1
= 2ax^2 + 2bx + 2c+1

but this is equal to 2x^2 + 4x + 1
so 2a = 2 , a = 1
2b = 4, b = 2
2c+1 = 1, c = 0

so g(x) = x^2 + 2x