To find f(x), we first need to express f in terms of g(x). We’re given that f(g(x)) = 3x² + 2x + 1. Knowing that g(x) = 2x + 2, we can substitute g(x) into the equation.
Let’s set y = g(x), which gives us: y = 2x + 2. We can solve for x in terms of y:
- Subtract 2 from both sides: y – 2 = 2x
- Now divide by 2: x = (y – 2)/2
Now we can substitute this expression for x back into f(g(x)):
f(g(x)) = f(y) = 3((y – 2)/2)² + 2((y – 2)/2) + 1
Now let’s simplify this:
- First, compute: 3((y – 2)/2)² = 3((y² – 4y + 4)/4) = (3y² – 12y + 12)/4
- Next, compute: 2((y – 2)/2) = y – 2
Putting it all together:
f(y) = (3y² – 12y + 12)/4 + (y – 2) + 1
Now let’s combine everything:
- f(y) = (3y² – 12y + 12)/4 + (4y – 8)/4 + 4/4
- f(y) = (3y² – 12y + 12 + 4y – 8 + 4)/4
- f(y) = (3y² – 8y + 8)/4
Finally, we can express f in terms of x (remember y = g(x)):
So the function becomes:
f(x) = (3x² – 8x + 8)/4
This is the function f(x) given the conditions provided in the problem.