let g(x)=2x+5 and h(x)=x^2-1, find g[h(x^3)]

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.

Learn how to compute g[h(x^3)] using function composition and polynomial functions. This problem involves substituting and simplifying functions g(x) = 2x + 5 and h(x) = x^2 - 1, resulting in the expression g[h(x^3)] = 2x^6 + 3.

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