If `f:RrarrR` be given by `f(x)=(3-x^(3))^(1//3)`, then `f^(-1)(x)` equals:

Options

A. `x^(3)`
B. `x^(1//3)`
C. `3-x^(3)`
D. `(3-x^(3))^(1//3)`

Answer

To find the inverse of the function \( f(x) = (3 - x^3)^{1/3} \), we will follow these steps:

### Step 1: Set the function equal to \( y \)
Let:
\[ y = f(x) = (3 - x^3)^{1/3} \]

### Step 2: Cube both sides to eliminate the cube root
Cubing both sides gives:
\[ y^3 = 3 - x^3 \]

### Step 3: Rearrange the equation to solve for \( x^3 \)
Rearranging the equation, we have:
\[ x^3 = 3 - y^3 \]

### Step 4: Solve for \( x \)
Taking the cube root of both sides, we find:
\[ x = (3 - y^3)^{1/3} \]

### Step 5: Replace \( y \) with \( x \) to find \( f^{-1}(x) \)
Since we want to express the inverse function in terms of \( x \), we replace \( y \) with \( x \):
\[ f^{-1}(x) = (3 - x^3)^{1/3} \]

Thus, the final answer is:
\[ f^{-1}(x) = (3 - x^3)^{1/3} \]

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