Let `f(x)=[x] +sqrt({x})`, where [.] denotes the integral part of x and {x} denotes the fractional part of x. Then `f^(-1)(x)` is

Options

A. `[x] +sqrt({x})`
B. `[x] + {x}^(2)`
C. `[x]^(2)+{x}`
D. `{x}+sqrt({x})`

Explanation

To find the inverse function \( f^{-1}(x) \) for the function \( f(x) = [x] + \sqrt{\{x\}} \), where \([x]\) denotes the integral part of \(x\) and \(\{x\}\) denotes the fractional part of \(x\), we can follow these steps:

### Step 1: Define the function
We start with the function:
\[ f(x) = [x] + \sqrt{\{x\}} \]
Here, \([x]\) is the greatest integer less than or equal to \(x\) and \(\{x\} = x - [x]\) is the fractional part of \(x\).

### Step 2: Set up the equation for the inverse
Let \(y = f(x)\):
\[ y = [x] + \sqrt{\{x\}} \]
We need to express \(x\) in terms of \(y\).

### Step 3: Separate the integer and fractional parts
From the equation \(y = [x] + \sqrt{\{x\}}\), we can denote:
\[ [x] = n \quad \text{(where \(n\) is an integer)} \]
Thus, we can rewrite \(y\) as:
\[ y = n + \sqrt{\{x\}} \]
This implies:
\[ \sqrt{\{x\}} = y - n \]
Squaring both sides gives:
\[ \{x\} = (y - n)^2 \]

### Step 4: Express \(x\) in terms of \(y\)
Since \(\{x\} = x - [x]\), we have:
\[ x = [x] + \{x\} = n + (y - n)^2 \]
So, we can express \(x\) as:
\[ x = n + (y - n)^2 \]

### Step 5: Determine \(n\)
Since \(n = [x]\), we know that \(n\) must satisfy:
\[ n \leq y < n + 1 \]
This implies:
\[ n = [y] \]

### Step 6: Substitute \(n\) back into the equation for \(x\)
Substituting \(n\) into the expression for \(x\):
\[ x = [y] + (y - [y])^2 \]
Thus, we have:
\[ f^{-1}(y) = [y] + (y - [y])^2 \]

### Final Result
Therefore, the inverse function is:
\[ f^{-1}(x) = [x] + (x - [x])^2 \]