Given the function f(2x+2)=-4x-2 and g(x+3)=6-3x

Question

Find:

a) f(x) & g(x)

b) (f°g)

Answer 1

First, notice that f°g = f(g(x)), so part b is pretty straight forward once you get part a. I'll show how to get f(x), and g(x) is exactly the same pattern.

You need to assume the FORM of f(x), and then COMPARE COEFFICIENTS of like terms.

f(x) is a regular polynomial - no logarithms or trig to worry about. You know this because you don't see any logs or sin/cos terms anywhere. Actually, it's a linear polynomial, which means no squares or higher powers.

1. Assume the form: Let f(t) = A + Bt + Ct² + Dt³ + ...
2. Remove all higher powers: f(t) = A + Bt
3. Put in the argument: f(2x+2) = A + B(2x+2)
4. Multiply and collect terms: = A + 2Bx +2B = 2Bx + (A + 2B)
5. Compare coefficients:
6. -4x - 2 === 2Bx + (A + 2B)
7. -4x = 2Bx -> B = -2
8. -2 = A + 2B = A - 4 -> A = 2
9. Put your coefficients back into the function you're building: f(x) = A + Bx = 2 - 2x
10. Check your work: f(2x + 2) = 2 - 2(2x + 2) = 2 - 4x - 4 = -4x - 2