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Let's break down the given problem step-by-step.

You are asked to use the functions \( f(x) = 5x - 2 \) and \( g(x) = 3 - x^2 \) to evaluate the following expressions:

### (a) \( (f \circ g)(-2) \)
This is the composition of functions, meaning you first evaluate \( g(-2) \) and then substitute that result into \( f(x) \).

1. **Evaluate \( g(-2) \):**
   \[
   g(-2) = 3 - (-2)^2 = 3 - 4 = -1
   \]

2. **Now, substitute \( g(-2) = -1 \) into \( f(x) \):**
   \[
   f(-1) = 5(-1) - 2 = -5 - 2 = -7
   \]

Thus, \( (f \circ g)(-2) = -7 \).

---

### (b) \( (g \circ f)(-2) \)
This is also a composition of functions, but this time you first evaluate \( f(-2) \) and then substitute that result into \( g(x) \).

1. **Evaluate \( f(-2) \):**
   \[
   f(-2) = 5(-2) - 2 = -10 - 2 = -12
   \]

2. **Now, substitute \( f(-2) = -12 \) into \( g(x) \):**
   \[
   g(-12) = 3 - (-12)^2 = 3 - 144 = -141
   \]

Thus, \( (g \circ f)(-2) = -141 \).

---

Let me know if you need any further details or if something was unclear.

Here are five related questions to expand on this:

1. What is the general formula for the composition of two functions?
2. How would you compute \( (f \circ g)(x) \) for any value of \( x \)?
3. Can you describe how function composition differs from regular function evaluation?
4. What are the domains of the composed functions \( (f \circ g)(x) \) and \( (g \circ f)(x) \)?
5. How would the result change if different values for \( x \) were chosen?

**Tip:** When working with function compositions, always evaluate the innermost function first!

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

Learn how to solve the function composition problem using f(x) = 5x - 2 and g(x) = 3 - x^2. Evaluate (f ∘ g)(-2) and (g ∘ f)(-2) with step-by-step solutions, covering core algebraic concepts suitable for Grades 9-12.

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