If `f(x) = [4-(x-7)^(3)],` then `f^(-1)(x)= ………… .`

Explanation

To find the inverse of the function \( f(x) = 4 - (x - 7)^3 \), we will follow these steps:

### Step 1: Set \( y = f(x) \)
Let \( y = f(x) \):
\[ y = 4 - (x - 7)^3 \]

### Step 2: Rearrange the equation to solve for \( x \)
We need to express \( x \) in terms of \( y \). Start by isolating the cube term:
\[ y - 4 = - (x - 7)^3 \]
Multiplying both sides by -1 gives:
\[ 4 - y = (x - 7)^3 \]

### Step 3: Take the cube root of both sides
Now, take the cube root of both sides:
\[ x - 7 = \sqrt[3]{4 - y} \]

### Step 4: Solve for \( x \)
Add 7 to both sides to solve for \( x \):
\[ x = 7 + \sqrt[3]{4 - y} \]

### Step 5: Replace \( y \) with \( x \) to find \( f^{-1}(x) \)
To express the inverse function, replace \( y \) with \( x \):
\[ f^{-1}(x) = 7 + \sqrt[3]{4 - x} \]

### Final Answer
Thus, the inverse function is:
\[ f^{-1}(x) = 7 + \sqrt[3]{4 - x} \]

---