To solve this problem, we need to evaluate the combined function $(f \circ g)(-2)$ for the given functions $f(x) = x^3 + x^2$ and $g(x) = x^3 - 4$.

Step-by-Step Solution

1. Evaluate $g(-2)$: Substitute $x = -2$ into $g(x) = x^3 - 4$: $g(-2) = (-2)^3 - 4 = -8 - 4 = -12$ So, $g(-2) = -12$.

2. Evaluate $f(g(-2)) = f(-12)$: Now we need to substitute $x = -12$ into $f(x) = x^3 + x^2$: $f(-12) = (-12)^3 + (-12)^2$ Calculate each term separately:

   • $(-12)^3 = -1728$
   • $(-12)^2 = 144$

   Therefore, $f(-12) = -1728 + 144 = -1584$

Final Answer

$(f \circ g)(-2) = -1584$

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

Here are five related questions for further practice:

1. How would you evaluate $(g \circ f)(-2)$ for the same functions?
2. What is $(f \circ g)(1)$ for the given functions?
3. How would you approach finding $(f \circ g)(0)$?
4. If $h(x) = x^2 + 3x$, how would you calculate $(f \circ h)(2)$?
5. What does it mean for a function to be "one-to-one," and how does this apply to $f(x)$ or $g(x)$?

Tip: When working with composition of functions, always start by evaluating the innermost function first, then substitute its result into the outer function.