https://file.aiquickdraw.com/mathbot/file/17262453563350qq96o05.jpg

The image shows two functions:

- \( f(x) = 5x - 2 \)
- \( g(x) = 3 - x^2 \)

You are asked to evaluate the following expressions:
1. \( f(f(2)) \)
2. \( g(g(3)) \)

### (a) \( f(f(2)) \)
First, calculate \( f(2) \):
\[
f(2) = 5(2) - 2 = 10 - 2 = 8
\]

Next, substitute \( 8 \) into \( f(x) \) again:
\[
f(f(2)) = f(8) = 5(8) - 2 = 40 - 2 = 38
\]

So, \( f(f(2)) = 38 \).

### (b) \( g(g(3)) \)
First, calculate \( g(3) \):
\[
g(3) = 3 - 3^2 = 3 - 9 = -6
\]

Next, substitute \( -6 \) into \( g(x) \):
\[
g(g(3)) = g(-6) = 3 - (-6)^2 = 3 - 36 = -33
\]

So, \( g(g(3)) = -33 \).

Let me know if you'd like more details or if you have any questions!

Here are some related questions to expand the discussion:
1. What is the domain and range of the function \( f(x) \)?
2. How would you find the inverse of \( g(x) = 3 - x^2 \)?
3. What happens if you compose \( g(f(x)) \) instead of \( f(g(x)) \)?
4. How would you approach solving \( f(f(x)) = 0 \)?
5. What is the effect of composing two quadratic functions?

**Tip:** When working with compositions of functions, it's important to solve them step by step and substitute carefully.

Use f(x) = 5x - 2 and g(x) = 3 - x^2 to evaluate the expressions f(f(2)) and g(g(3)).

Learn how to evaluate composed functions, including f(f(2)) and g(g(3)), with step-by-step calculations. Involves linear and quadratic functions, making it suitable for Grades 9-11.

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