https://file.aiquickdraw.com/mathbot/file/1727047021922prc0kd98.jpg

Let's go through each of the sub-problems step by step.

We are given two functions:
\[
f(x) = x^2 - 2 \quad \text{and} \quad g(x) = 3x - 2
\]

### Part a: \( f(-2) \)
We substitute \( x = -2 \) into the function \( f(x) \):

\[
f(-2) = (-2)^2 - 2 = 4 - 2 = 2
\]

Thus, \( f(-2) = 2 \).

### Part b: \( f(-1) + g(-2) \)
1. First, calculate \( f(-1) \):
\[
f(-1) = (-1)^2 - 2 = 1 - 2 = -1
\]

2. Now, calculate \( g(-2) \):
\[
g(-2) = 3(-2) - 2 = -6 - 2 = -8
\]

Now, add the two results:
\[
f(-1) + g(-2) = -1 + (-8) = -9
\]

Thus, \( f(-1) + g(-2) = -9 \).

### Part c: \( 3f(x) + g(x) \)
We substitute the expressions for \( f(x) \) and \( g(x) \):

\[
3f(x) + g(x) = 3(x^2 - 2) + (3x - 2)
\]

Now, expand and simplify:

\[
3f(x) + g(x) = 3x^2 - 6 + 3x - 2 = 3x^2 + 3x - 8
\]

Thus, \( 3f(x) + g(x) = 3x^2 + 3x - 8 \).

### Part d: \( f(g(4)) \)
1. First, calculate \( g(4) \):
\[
g(4) = 3(4) - 2 = 12 - 2 = 10
\]

2. Now, substitute this result into \( f(x) \):
\[
f(10) = (10)^2 - 2 = 100 - 2 = 98
\]

Thus, \( f(g(4)) = 98 \).

### Part e: \( g(f(4)) \)
1. First, calculate \( f(4) \):
\[
f(4) = (4)^2 - 2 = 16 - 2 = 14
\]

2. Now, substitute this result into \( g(x) \):
\[
g(14) = 3(14) - 2 = 42 - 2 = 40
\]

Thus, \( g(f(4)) = 40 \).

### Final Answers:
- a. \( f(-2) = 2 \)
- b. \( f(-1) + g(-2) = -9 \)
- c. \( 3f(x) + g(x) = 3x^2 + 3x - 8 \)
- d. \( f(g(4)) = 98 \)
- e. \( g(f(4)) = 40 \)

Would you like any further details or explanations?

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Here are five related questions for further practice:
1. What is \( f(g(-1)) \)?
2. Find the value of \( g(f(3)) \).
3. Simplify \( 2f(x) + g(x) \).
4. What is \( f(x) + g(x) \) in general form?
5. Calculate \( f(g(0)) \).

**Tip:** When evaluating composite functions, always start from the inside and work outward!

Let f(x) = x^2 - 2 and g(x) = 3x - 2, evaluate the following: a) f(-2) b) f(-1) + g(-2) c) 3f(x) + g(x) d) f(g(4)) e) g(f(4))

Learn how to evaluate functions and their composites with the given problem: f(x) = x^2 - 2 and g(x) = 3x - 2. This exercise covers function evaluation, including substitution and simplification. Suitable for Grades 9-11.

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