To find the function f(x), we need to start from the given equation: f(g(x)) = 3x² + 2x + 1 where g(x) = 2x + 2. This means we need to express the input of f in terms of x using g.
First, let’s define g(x):
- g(x) = 2x + 2
Next, we will solve for x in terms of g(x):
- Let y = g(x) so y = 2x + 2.
- To express x in terms of y, we can rearrange this to:
- y – 2 = 2x
- x = (y – 2)/2
Now we substitute x = (y – 2)/2 into the equation for f(g(x)):
- Substituting x back gives:
- f(y) = 3((y – 2)/2)² + 2((y – 2)/2) + 1
Next, let’s expand the equation:
- First, calculate 3((y – 2)/2)²:
- 3((y² – 4y + 4)/4) = (3y² – 12y + 12)/4
- Next, calculate 2((y – 2)/2):
- 2(y – 2)/2 = y – 2
Combining the expanded terms:
- f(y) = (3y² – 12y + 12)/4 + y – 2 + 1
- To combine these, simplify further:
- 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
Thus, we have derived that:
- f(y) = rac{3y² – 8y + 8}{4}
Now replace y back with g(x):
- f(x) = rac{3(2x+2)² – 8(2x+2) + 8}{4}
So we can conclude that:
- f(x) = rac{3(4x² + 8x + 4) – (16x + 16) + 8}{4}
- Finally, simplify this to find a clear view of f(x):
f(x) = 3x² + 4x + 2
Therefore, the function f(x) is:
f(x) = 3x² + 4x + 2.