f(x)=
−3x−10
g, of, x, equals, 3, x, squared, minus, 6, x, minus, 12
g(x)=
3x 
2
 −6x−12
Find: , f, of, g, of, x
Find: f(g(x))

We are asked to find \( f(g(x)) \) where:

- \( f(x) = -3x - 10 \)
- \( g(x) = 3x^2 - 6x - 12 \)

To find \( f(g(x)) \), we substitute \( g(x) \) into \( f(x) \). That is, we replace \( x \) in \( f(x) \) with \( g(x) \).

### Steps:

1. Substitute \( g(x) = 3x^2 - 6x - 12 \) into \( f(x) = -3x - 10 \).
   \[
   f(g(x)) = -3(3x^2 - 6x - 12) - 10
   \]

2. Distribute the \( -3 \) across the terms inside the parentheses:
   \[
   f(g(x)) = -9x^2 + 18x + 36 - 10
   \]

3. Simplify the constant terms:
   \[
   f(g(x)) = -9x^2 + 18x + 26
   \]

Thus, the function \( f(g(x)) \) is:
\[
f(g(x)) = -9x^2 + 18x + 26
\]

Would you like further explanation or details on any step?

Here are 5 related questions you can explore:
1. What is \( g(f(x)) \) for the same functions?
2. How do you find the domain of \( f(g(x)) \)?
3. What happens to the graph of \( f(x) \) when composed with \( g(x) \)?
4. Can you express \( f(g(x)) \) in factored form?
5. How would the composition change if \( g(x) = 3x^2 + 6x - 12 \)?

**Tip**: When composing functions, always replace the input variable with the entire second function carefully.

Find f(g(x)) where f(x) = -3x - 10 and g(x) = 3x^2 - 6x - 12.

Learn how to find the composition of functions f(x) = -3x - 10 and g(x) = 3x^2 - 6x - 12. This solution involves algebraic manipulation and quadratic functions, providing a step-by-step approach to calculate f(g(x)). Suitable for students in Grades 9-11.

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