Let `f:DtoR`, where D is the domain of `f`. Find the inverse of `f` if it exists: `f(x)=(4-(x-7)^(3))^(1//5)`

Explanation

To find the inverse of the function \( f(x) = (4 - (x - 7)^3)^{1/5} \), we will follow these steps:

### Step 1: Check if the function is one-to-one
To determine if \( f \) is one-to-one, we assume \( f(x_1) = f(x_2) \) for \( x_1, x_2 \in D \).

\[ f(x_1) = f(x_2) \implies (4 - (x_1 - 7)^3)^{1/5} = (4 - (x_2 - 7)^3)^{1/5} \]

Raising both sides to the power of 5:

\[ 4 - (x_1 - 7)^3 = 4 - (x_2 - 7)^3 \]

Subtracting 4 from both sides:

\[ -(x_1 - 7)^3 = -(x_2 - 7)^3 \]

This simplifies to:

\[ (x_1 - 7)^3 = (x_2 - 7)^3 \]

Taking the cube root of both sides:

\[ x_1 - 7 = x_2 - 7 \]

Thus, we have:

\[ x_1 = x_2 \]

Since \( x_1 = x_2 \), the function \( f \) is one-to-one.

### Step 2: Check if the function is onto
To check if \( f \) is onto, we need to express \( x \) in terms of \( y \). We set \( y = f(x) \):

\[ y = (4 - (x - 7)^3)^{1/5} \]

Raising both sides to the power of 5:

\[ y^5 = 4 - (x - 7)^3 \]

Rearranging gives:

\[ (x - 7)^3 = 4 - y^5 \]

Taking the cube root:

\[ x - 7 = \sqrt[3]{4 - y^5} \]

Adding 7 to both sides:

\[ x = 7 + \sqrt[3]{4 - y^5} \]

Since for every value of \( y \) there is a corresponding value of \( x \), the function \( f \) is onto.

### Step 3: Find the inverse function
Now that we have established that \( f \) is both one-to-one and onto, we can find the inverse function. We replace \( y \) with \( x \) and \( x \) with \( f^{-1}(x) \):

\[ f^{-1}(x) = 7 + \sqrt[3]{4 - x^5} \]

### Final Result
Thus, the inverse function is:

\[ f^{-1}(x) = 7 + \sqrt[3]{4 - x^5} \]