To find f(g(x)), we need to substitute g(x) into the function f(x).
First, let’s rewrite the functions clearly:
f(x) = 2x² + 2x + 7
g(x) = x² + 9x + 6
Now we can substitute g(x) into f(x):
f(g(x)) = f(x² + 9x + 6)
We replace every x in f(x) with (x² + 9x + 6):
f(g(x)) = 2((x² + 9x + 6)²) + 2(x² + 9x + 6) + 7
Now we need to simplify this further. First, calculate (x² + 9x + 6)²:
(x² + 9x + 6)² = x^4 + 18x^3 + 117x^2 + 108x + 36 (using the formula (a + b + c)²)
Substituting back into f(g(x)):
f(g(x)) = 2(x^4 + 18x^3 + 117x^2 + 108x + 36) + 2(x² + 9x + 6) + 7
Now, distribute the 2:
f(g(x)) = 2x^4 + 36x^3 + 234x^2 + 216x + 72 + 2x² + 18x + 12 + 7
Combine like terms:
f(g(x)) = 2x^4 + 36x^3 + 236x^2 + 234x + 91
Thus, f(g(x)) simplifies to 2x^4 + 36x^3 + 236x^2 + 234x + 91.