linear functions question. (please help me, my brain has been raked)?

Question

1. if F(x)= -3x + 2, what are the values of m and b so that
F(mx+b)= 2x - 3?

2. Let G(x)= 2x² - 4. if (F○G)(x)= 2x-4, find F(x).

3. give two linear functions F(x) and G(x) such that F(G(x))= 3x - 2.

Please include a detailed solution to the problem. I know I'm asking much, but please help me.
Thank You!

Answer 1

#1 - substitute mx+b for x in the first equation, simplify, then let the coefficeint of x=2 and solve for m and the constant equal -3 and solve for b. Thus:
-3(mx+b)+2
-3mx + 2-3b
-3m=2, 2-3b=-3
m=-2/3, b=5/3

#2 In this case, you need to find a series of algebraic manipulations that transforms 2x²-4 into 2x-4. To do this, note that 2x²-4 factors into 2(x-√2)(x+√2). Divide by x+√2 to get 2x-2√2, then add 2√2 and subtract 4. Thus, f(x)=x/(x+√2)+2√2-4.

#3 This one's trivial. G(x) = 3x and F(x)=x-2

Answer 2

#1 - F(mx+b) = -3(mx+b) +2 = -3mx -3b +2
m is quickly done. -3m = 2 so m = -2/3
-3b+ 2 = -3, so b=5/3

#2 - What if FOG, is this a weather question?

#3 F(x) = x
G(x) = 3x -2

Answer 3

#2.
let t(x) = 2x – 4
fog(x) = t(x) => (fog)^(-1) (x) = t^(-1) (x)
=> g^(-1) o f^(-1) (x) = t^(-1) (x)

t^(-1) (x) = (x+4) / 2
g^(-1) (x) = ( (x+4)/ 2 ) ^ (1/2)

=> ( (f^(-1) (x) + 4) / 2)^(1/2) = (x+4) / 2
=> 2 ( (f^(-1) (x) + 4) / 2)^(1/2) – 4 = x
=> (2 (f^(-1) (x) + 4) )^(1/2) -4 = x
=> f(x) = (2(x + 4))^(1/2) – 4