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The function provided is:

\[
f(x) = \sqrt[3]{3x - 2}
\]

### Step 1: Find the inverse function

To find the inverse of a function \( f(x) \), follow these steps:

1. Replace \( f(x) \) with \( y \):
   \[
   y = \sqrt[3]{3x - 2}
   \]

2. Swap \( x \) and \( y \) (this reflects finding the inverse):
   \[
   x = \sqrt[3]{3y - 2}
   \]

3. Now, solve for \( y \):
   - Cube both sides to remove the cube root:
     \[
     x^3 = 3y - 2
     \]
   - Isolate \( y \):
     \[
     3y = x^3 + 2
     \]
     \[
     y = \frac{x^3 + 2}{3}
     \]

So, the inverse function is:

\[
f^{-1}(x) = \frac{x^3 + 2}{3}
\]

### Step 2: Prove the inverse by composition

To verify that \( f^{-1}(x) \) is the correct inverse, we need to check two conditions:

1. \( f(f^{-1}(x)) = x \)
2. \( f^{-1}(f(x)) = x \)

#### 1. Prove \( f(f^{-1}(x)) = x \):
Substitute \( f^{-1}(x) = \frac{x^3 + 2}{3} \) into \( f(x) = \sqrt[3]{3x - 2} \):
\[
f(f^{-1}(x)) = f\left( \frac{x^3 + 2}{3} \right) = \sqrt[3]{3 \left( \frac{x^3 + 2}{3} \right) - 2}
\]
Simplify the expression inside the cube root:
\[
f(f^{-1}(x)) = \sqrt[3]{x^3 + 2 - 2} = \sqrt[3]{x^3} = x
\]
This shows that \( f(f^{-1}(x)) = x \).

#### 2. Prove \( f^{-1}(f(x)) = x \):
Substitute \( f(x) = \sqrt[3]{3x - 2} \) into \( f^{-1}(x) = \frac{x^3 + 2}{3} \):
\[
f^{-1}(f(x)) = f^{-1}\left( \sqrt[3]{3x - 2} \right) = \frac{\left( \sqrt[3]{3x - 2} \right)^3 + 2}{3}
\]
Simplify the expression:
\[
f^{-1}(f(x)) = \frac{3x - 2 + 2}{3} = \frac{3x}{3} = x
\]
This shows that \( f^{-1}(f(x)) = x \).

### Conclusion:
Since both compositions result in \( x \), we have verified that the inverse function is correct:

\[
f^{-1}(x) = \frac{x^3 + 2}{3}
\]

Let me know if you would like further clarifications or details. Here are some related questions:

1. How do you find the inverse of more complex functions?
2. What is the domain and range of the inverse function?
3. Can you explain why the domain of \( f(x) \) is all real numbers?
4. How does composition help in verifying inverse functions?
5. What other methods exist for verifying inverse functions?

**Tip:** When solving for the inverse of a function, always check if the function is one-to-one, as this ensures that it has an inverse.

Find the inverse of the function f(x) given, then prove (by composition) your inverse function is correct. Note the domain of f is all real numbers. f(x) = ∛(3x - 2)

Learn how to find the inverse of the function f(x) = ∛(3x - 2) and prove the correctness of the inverse using function composition. This solution includes step-by-step explanations and covers key algebraic concepts, including cubic roots and function inverses. Suitable for high school students in Grades 10-12.

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