Express your answer as a polynomial in standard form.

f, of, x, equals, minus, 3, x, plus, 12

*f*(*x*)=

−3*x*+12

g, of, x, equals, 3, x, squared, minus, 6, x, minus, 6

*g*(*x*)=

3*x*2−6*x*−6

Find: , left parenthesis, f, circle, g, right parenthesis, left parenthesis, x, right parenthesis

Find: (*f*∘*g*)(*x*)

To find \((f \circ g)(x)\), we need to compute the composition of functions \(f(x)\) and \(g(x)\), which means substituting \(g(x)\) into \(f(x)\).

Given:
- \( f(x) = -3x + 12 \)
- \( g(x) = 3x^2 - 6x - 6 \)

To find \((f \circ g)(x)\), we replace \(x\) in \(f(x)\) with \(g(x)\). So we have:
\[
(f \circ g)(x) = f(g(x)) = f(3x^2 - 6x - 6)
\]

Substitute \(g(x)\) into \(f(x)\):
\[
f(g(x)) = -3(3x^2 - 6x - 6) + 12
\]

Now distribute the \(-3\):
\[
f(g(x)) = -9x^2 + 18x + 18 + 12
\]

Combine like terms:
\[
f(g(x)) = -9x^2 + 18x + 30
\]

Thus, the composition \((f \circ g)(x)\) is:
\[
(f \circ g)(x) = -9x^2 + 18x + 30
\]

Would you like more details or have any questions about this? Here are five related questions to explore further:

1. How do you find the composition of functions in general?
2. What is the difference between \((f \circ g)(x)\) and \((g \circ f)(x)\)?
3. Can you determine the degree of a polynomial after composition?
4. What happens when both functions are quadratic, not linear like \(f(x)\)?
5. How can we determine if the composition of two functions is one-to-one?

**Tip:** Always simplify the expression after substitution to make sure the function is in standard polynomial form.

Find: (f∘g)(x) given f(x) = -3x + 12 and g(x) = 3x^2 - 6x - 6. Express your answer as a polynomial in standard form.

Learn how to find the composition of functions (f∘g)(x) with a linear function f(x) and a quadratic function g(x). This problem demonstrates the process of substituting g(x) into f(x) and simplifying to express the result as a polynomial in standard form. Suitable for students in Grades 10-12.

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