What is the function f(x) if f(g(x)) = 3x² + 2x + 1 and g(x) = 2x + 2?

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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.

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