*f*(*x*)=

−2*x*−10

g, of, x, equals, x, squared, plus, 5, x, plus, 9

*g*(*x*)=

*x*2+5*x*+9

Find: , g, of, f, of, x

Find: *g*(*f*(*x*))

To find \( g(f(x)) \), we need to substitute the function \( f(x) \) into the function \( g(x) \).

Given:
- \( f(x) = -2x - 10 \)
- \( g(x) = x^2 + 5x + 9 \)

### Steps:
1. First, substitute \( f(x) = -2x - 10 \) into \( g(x) \):
   \[
   g(f(x)) = g(-2x - 10)
   \]

2. Now substitute \( -2x - 10 \) for \( x \) in the expression for \( g(x) \):
   \[
   g(-2x - 10) = (-2x - 10)^2 + 5(-2x - 10) + 9
   \]

3. Expand the terms:
   \[
   (-2x - 10)^2 = (4x^2 + 40x + 100)
   \]
   \[
   5(-2x - 10) = -10x - 50
   \]

4. Now, substitute these expanded values back into the equation:
   \[
   g(-2x - 10) = 4x^2 + 40x + 100 - 10x - 50 + 9
   \]

5. Combine like terms:
   \[
   g(-2x - 10) = 4x^2 + (40x - 10x) + (100 - 50 + 9)
   \]
   \[
   g(-2x - 10) = 4x^2 + 30x + 59
   \]

Thus, \( g(f(x)) = 4x^2 + 30x + 59 \).

Would you like further details or explanations? 

Here are some related questions to explore:
1. How do you compute \( g(f(x)) \) for different types of functions?
2. What are the steps for finding \( f(g(x)) \)?
3. How do transformations affect the composition of functions?
4. Can \( g(f(x)) \) be used to model real-world scenarios?
5. How would \( g(f(x)) \) change if \( f(x) \) were a different linear function?

**Tip:** Always expand terms carefully in compositions to avoid mistakes in signs or coefficients!

Find g(f(x)) where f(x) = -2x - 10 and g(x) = x^2 + 5x + 9.

Learn how to solve function compositions by finding g(f(x)) where f(x) is linear and g(x) is quadratic. This step-by-step guide explains how to substitute and expand terms to simplify the expression, making it suitable for students in Grades 9-12.

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