If `f(x)= {4 - (x-7)^(3)}`, then find `f^(-1)(x)=`

Options

A. `(4-x)^(1/3)+7`
B. `(4-x)^((1)/(3))-7`
C. `(4+x)^(3)+7`
D. None of these

Answer

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

### Step 1: Rewrite the function
We start with the function:
\[ f(x) = 4 - (x - 7)^3 \]
We will set \( f(y) = x \) instead of \( f(x) \) to find the inverse.

### Step 2: Set up the equation
Now, we can write:
\[ x = 4 - (y - 7)^3 \]

### Step 3: Isolate the cubic term
To isolate the cubic term, we rearrange the equation:
\[ (y - 7)^3 = 4 - x \]

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

### Step 5: Solve for \( y \)
Now, we solve for \( y \) by adding 7 to both sides:
\[ y = 7 + \sqrt[3]{4 - x} \]

### Step 6: Write the inverse function
Since we set \( f(y) = x \), we can express the inverse function as:
\[ f^{-1}(x) = 7 + \sqrt[3]{4 - x} \]

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