Final answer:
To calculate the composite function (f g)(x), apply g(x) to x to get f(g(x)). Then, use the function f(x) = (x²)² on the result of g(x), which is (2 - 3x). Finally, square the result (2 - 3x)² and then square this outcome again.
Step-by-step explanation:
To find the composite function (f g)(x), first you must apply the function g(x) to x and then apply the function f(x) to the result of g(x). The function g(x) = 2 - 3x when plugged into f(x) gives f(2 - 3x).
Now, since f(x) = (x²)², applying this to g(x) means we need to square 2 - 3x, and then square the result again. So we have ((2 - 3x)²)². Computing this step by step:
- Square 2-3x: (2 - 3x)² = 4 - 12x + 9x².
- Square the result of step 1: (4 - 12x + 9x²)².
The complete expansion of this expression can be quite lengthy and is not essential to understand the process. What is important is that you know you need to apply the exponent twice due to the composition of the functions.