Find the inverse of the function: `f:Z to Z` defined by `f(x)=[x+1],` where [.] denotes the greatest integer function.

Explanation

To find the inverse of the function \( f: \mathbb{Z} \to \mathbb{Z} \) defined by \( f(x) = [x + 1] \), where \([.]\) denotes the greatest integer function (also known as the floor function), we will follow these steps:

### Step 1: Understand the function
The function \( f(x) = [x + 1] \) means that for any integer \( x \), we take \( x + 1 \) and then apply the greatest integer function. Since \( x \) is an integer, \( [x + 1] \) is simply \( x + 1 \).

### Step 2: Express the function
We can express the function as:
\[ f(x) = x + 1 \]
for all \( x \in \mathbb{Z} \).

### Step 3: Set up the equation for the inverse
To find the inverse function, we set \( f(x) = y \). Thus, we have:
\[ y = x + 1 \]

### Step 4: Solve for \( x \)
Now, we need to solve for \( x \) in terms of \( y \):
\[ x = y - 1 \]

### Step 5: Write the inverse function
Since \( x = f^{-1}(y) \), we can write:
\[ f^{-1}(y) = y - 1 \]

### Step 6: Replace \( y \) with \( x \)
To express the inverse function in standard form, we replace \( y \) with \( x \):
\[ f^{-1}(x) = x - 1 \]

### Final Answer
Thus, the inverse of the function \( f \) is:
\[ f^{-1}(x) = x - 1 \]

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