Composite funtions

Question

Given:

g(x)=x^2+x-1, 

(fOg)(x)=x^4+2x^3-3x^2-4x+6

what does f(x)=___________________?

Answer 1

Since f(g(x)) will need to produce an x⁴ term, f(x) must have an x² term (because (x²)² = x⁴. Based on that, lets figure out what the result of that first term would be if we square g(x).

(g(x))² = (x² + x - 1)² = x⁴ + 2x³ - x² - 2x + 1

That gives us both the first and second terms of f(g(x)) so let's look at the third term. I would need to have an additional -2x² to get the correct third term of f(g(x)) meaning I must need to multiply g(x) by -2.

-2(g(x)) = -2(x² + x - 1) = -2x² - 2x + 2

Adding that to the result of squaring g(x) = (x⁴ + 2x³ - x² - 2x + 1) + (-2x² - 2x + 2) or:

x⁴ + 2x³ - 3x² - 4x + 3

That is everything except I need to add 3 more. So f(x) = x² - 2x + 3