Let's solve each part step by step:

1. $f(-3)$

The function $f(x) = x^2 - 4$. To find $f(-3)$, substitute $x = -3$ into the function: $f(-3) = (-3)^2 - 4 = 9 - 4 = 5$ So, $f(-3) = 5$.

2. $f(-3) + g(x)$

We already know $f(-3) = 5$. Now, we need to find the expression for $g(x)$. The function $g(x) = 3x + 2$. So, $f(-3) + g(x) = 5 + (3x + 2) = 5 + 3x + 2 = 7 + 3x$ Thus, $f(-3) + g(x) = 7 + 3x$.

3. $f(g(3))$

We need to find $g(3)$ first. Using $g(x) = 3x + 2$: $g(3) = 3(3) + 2 = 9 + 2 = 11$ Now, substitute $g(3) = 11$ into $f(x)$: $f(g(3)) = f(11) = 11^2 - 4 = 121 - 4 = 117$ Thus, $f(g(3)) = 117$.

4. $g(f(3))$

We need to find $f(3)$ first. Using $f(x) = x^2 - 4$: $f(3) = 3^2 - 4 = 9 - 4 = 5$ Now, substitute $f(3) = 5$ into $g(x)$: $g(f(3)) = g(5) = 3(5) + 2 = 15 + 2 = 17$ Thus, $g(f(3)) = 17$.

Final Answers:

• $f(-3) = 5$
• $f(-3) + g(x) = 7 + 3x$
• $f(g(3)) = 117$
• $g(f(3)) = 17$

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Would you like further details on any of the steps? Or do you have any questions?

Here are some related questions you might find interesting:

1. What is the general form for composing two functions like $f(g(x))$?
2. How do you find the inverse of a function like $g(x) = 3x + 2$?
3. How do you solve a quadratic function for $x$?
4. What is the difference between evaluating a function and composing two functions?
5. How does function composition relate to real-world problems?

Tip: When composing functions, always evaluate the "inner" function first, then apply the result to the "outer" function.