To analyze Orville's work and the possible conclusions, let's first review the process for determining if two functions are inverses of each other.

Two functions $f(x)$ and $g(x)$ are inverses if and only if:

1. $f(g(x)) = x$ for all $x$ in the domain of $g$,
2. $g(f(x)) = x$ for all $x$ in the domain of $f$.

Orville computed $f(g(x))$, so let's check that process:

Step 1: Substituting $g(x)$ into $f(x)$

• Given: $f(x) = 4\sqrt[3]{x} - 2$ and $g(x) = (x + 24)^3$
• Substituting $g(x) = (x + 24)^3$ into $f(x)$: $f(g(x)) = f((x + 24)^3) = 4\sqrt[3]{(x + 24)^3} - 2$
• Simplifying: $4\sqrt[3]{(x + 24)^3} = 4(x + 24)$ $f(g(x)) = 4(x + 24) - 2 = 4x + 96 - 2 = 4x + 94$

This is not equal to $x$, but rather $4x + 94$. This means $f(g(x)) \neq x$.

Step 2: Conclusion from Orville’s Work

• Orville claimed that $f(g(x)) = x$, but we have shown that $f(g(x)) = 4x + 94$, so his work is incorrect.
• Since $f(g(x)) \neq x$, the two functions are not inverses.

Correct answers:

• B: Orville’s work is not correct.
• D: Orville’s work indicates that the functions are not inverses.
• F: The two functions $f$ and $g$ are not inverses.

Final Answer:

The three correct answers are:

• B, D, and F.

Do you want further details or explanations on any of the steps?

5 Related Questions:

1. How can we verify if two functions are inverses using their compositions?
2. What are common mistakes when checking if two functions are inverses?
3. How would you compute $g(f(x))$ in this case?
4. What are the domain restrictions when dealing with function inverses?
5. Can two functions be inverses if only one of the compositions equals $x$?

Tip:

When checking if two functions are inverses, always check both compositions, $f(g(x))$ and $g(f(x))$.