To analyze Orville's work and determine which of the provided options are correct, we need to check whether his composition of functions $f(x)$ and $g(x)$ is valid, and if the two functions are indeed inverses.

The functions are given as:

• $f(x) = \sqrt[3]{4x} - 2$
• $g(x) = (x + 24)^3$

Step-by-Step Analysis of Orville's Work:

1. Composition $f \circ g$: $f(g(x)) = f((x + 24)^3)$ Now, apply the function $f(x)$ to $g(x)$. Substituting $g(x) = (x + 24)^3$ into $f(x)$: $f(g(x)) = \sqrt[3]{4((x + 24)^3)} - 2$

2. Simplification of Orville's work: Orville proceeds with: $f(g(x)) = 4(x+24)^3 - 2$ which appears to be an incorrect manipulation. Specifically, the cube root operation has been incorrectly dropped.

3. Correct Approach: The correct approach would involve keeping the cube root when simplifying. There seems to be a mistake in how the cube root and power operations were handled. If we continue with Orville’s work as it stands, the conclusion $f \circ g(x) = x$ might not hold, as there was an algebraic error in the simplification.

Conclusion:

Based on this, the following three correct statements are:

• (B) Orville’s work is not correct. There is an algebraic mistake in simplifying the composition of the two functions.

• (D) Orville’s work indicates that the functions are not inverses. Due to the error in his work, the conclusion that the functions are inverses is incorrect.

• (F) The two functions $f(x)$ and $g(x)$ are not inverses. Since Orville made a mistake in his composition, the functions are not inverses.

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Would you like further details on this or have any other questions?

Related Questions:

1. How do you correctly compose two functions to check if they are inverses?
2. What are common mistakes when dealing with function composition?
3. How can you check whether two functions are inverses without explicitly composing them?
4. What happens if a function and its inverse are composed?
5. Can two functions that pass the horizontal line test still fail to be inverses?

Tip:

When checking if two functions are inverses, you need to check both $f(g(x)) = x$ and $g(f(x)) = x$ to ensure correctness.