Orville composed functions f(x)=4x−−√3−2f(x)=4x3−2 and g(x)=(x+24)3g(x)=(x+24)3 to see if the two functions are inverses of each other. His work is shown below.

f∘g=f(g(x))f∘g=f(g(x))
f∘g=f((x+24)3)f∘g=f((x+24)3)
f∘g=4(x+24)3−−−−−−√3−2f∘g=4(x+24)33−2
f∘g=4(x+24)−2f∘g=4(x+24)−2
f∘g=(x+2)−2f∘g=(x+2)−2
f∘g=xf∘g=x

Which of the following is true about Orville’s work?

Select **THREE** correct answers.

A.Orville’s work is correct.

B.Orville’s work is not correct.

C.Orville’s work indicates that the functions are inverses.

D.Orville’s work indicates that the functions are not inverses.

E.The two functions ff and gg are inverses

F.The two functions ff and gg are not inverses.

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 \( g(f(x)) \).

Orville composed functions f(x) = 4√3x−2 and g(x) = (x + 24)^3 to see if the two functions are inverses of each other. His work is shown below. Which of the following is true about Orville’s work?
Select THREE correct answers.

Learn how to verify if two composed functions f(x) = 4√3x−2 and g(x) = (x + 24)^3 are inverses of each other. This problem covers function composition and inverse function theorem concepts, suitable for students in Grades 9-11.

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