. Let g(x) = 3x−2 and h(x) = 9x2 +3x+4. Find a function f such that f ◦g = h. What is g ◦ f?

Answer 1

I suppose: f(x) = a X² + bX + c

Because h(x) has the term 9X² which tell me (3x-2) must reach to the power of 2.

(fog)(x) = a(3x-2)² +b(3x-2)+c → (fog)(x) = a(9x² - 12x + 4) + 3bx - 2b + c = 9ax² - (12a - 3b)x + (4a -2b +c)

We know that: (fog)(x) = h(x) → 9ax² - (12a - 3b)x + (4a -2b +c) = 9x² + 3x + 4 Now we can set 3 equations for our 3 unknowns (a,b,c).

9a = 9 (coefficient of x²) , - 12a + 3b = 3 (coefficient of x from both sides) , and 4a -2b +c = 4

9a = 9 → a = 1 , - 12a + 3b = 3 → -12 (1) + 3b = 3 → b = 5

4a -2b +c = 4 → 4(1) -2(5) + c = 4 → c = 10

f(x) = a X² + bX + c → f(x) = X² + 5X + 10

(gof)(x) = 3(X² + 5X + 10) - 2