To find $g[h(x^3)]$, we need to substitute $h(x^3)$ into $g(x)$.

Step 1: Compute $h(x^3)$

Given $h(x) = x^2 - 1$, we substitute $x^3$ into $h(x)$:

$h(x^3) = (x^3)^2 - 1 = x^6 - 1$

Step 2: Substitute $h(x^3)$ into $g(x)$

Now that we have $h(x^3) = x^6 - 1$, we substitute this into $g(x)$ where $g(x) = 2x + 5$:

$g[h(x^3)] = g(x^6 - 1)$

Substitute $x^6 - 1$ into $g(x)$:

$g(x^6 - 1) = 2(x^6 - 1) + 5$

Step 3: Simplify the expression

Now, simplify the expression:

$g(x^6 - 1) = 2x^6 - 2 + 5 = 2x^6 + 3$

Final Answer:

$g[h(x^3)] = 2x^6 + 3$

Would you like more details or have any questions?

Here are 8 related questions you might consider:

1. How do you find the composition of two functions in general?
2. What is the domain of the composed function $g[h(x)]$?
3. Can you reverse the composition and find $h[g(x)]$?
4. What would $g[h(x^2)]$ look like instead?
5. How do you determine if a function is one-to-one?
6. What is the significance of the result $g[h(x^3)] = 2x^6 + 3$?
7. How does composition of functions relate to function transformations?
8. What happens if you compose three or more functions?

Tip: When dealing with function composition, always ensure that you substitute correctly and simplify the expression thoroughly.