https://file.aiquickdraw.com/mathbot/file/1726578444874g4caor94.jpg

The image shows functions \( f(u) = 5u + 2 \) and \( g(u) = -u^2 \), and you are asked to perform some function compositions:

### a. \( f \circ g(u) \) 
This represents the composition of \( f(u) \) with \( g(u) \), meaning you will substitute \( g(u) \) into \( f(u) \).

Given:
- \( f(u) = 5u + 2 \)
- \( g(u) = -u^2 \)

To find \( f(g(u)) \):
\[
f(g(u)) = f(-u^2) = 5(-u^2) + 2 = -5u^2 + 2
\]

### b. \( g \circ f(u) \)
This is the composition of \( g(u) \) with \( f(u) \), meaning you will substitute \( f(u) \) into \( g(u) \).

To find \( g(f(u)) \):
\[
g(f(u)) = g(5u + 2) = -(5u + 2)^2 = -(25u^2 + 20u + 4) = -25u^2 - 20u - 4
\]

### c. \( g \circ g(u) \)
This is the composition of \( g(u) \) with itself, meaning you will substitute \( g(u) \) into \( g(u) \).

To find \( g(g(u)) \):
\[
g(g(u)) = g(-u^2) = -(-u^2)^2 = -u^4
\]

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Would you like more details or have any questions? Here are 5 related questions to expand your understanding:

1. What is the general formula for composing two functions?
2. How would the composition of functions differ if the domain of \( g(u) \) were restricted?
3. How does composition of functions relate to transformations in geometry?
4. What happens if you switch the order of composition between two different functions?
5. Can all functions be composed, or are there certain conditions?

**Tip:** When composing functions, always substitute the second function into the first, keeping track of any changes to variables carefully.

Given f(u) = 5u + 2 and g(u) = -u^2, find the compositions f(g(u)), g(f(u)), and g(g(u)).

Learn how to compose functions with step-by-step solutions for f(g(u)), g(f(u)), and g(g(u)). This problem covers algebraic concepts suitable for high school students, including function composition and quadratic functions.

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