Find the range of the following functions: (where {.} and [.] represent fractional part and greatest integer part functions respectively)
`f(x)=1-|x-2|`

Explanation

To find the range of the function \( f(x) = 1 - |x - 2| \), we will analyze the function step by step.

### Step 1: Understand the Absolute Value Function
The absolute value function \( |x - 2| \) can be defined piecewise:
- For \( x \geq 2 \), \( |x - 2| = x - 2 \)
- For \( x < 2 \), \( |x - 2| = -(x - 2) = 2 - x \)

### Step 2: Define the Function Piecewise
Using the definition of the absolute value, we can express \( f(x) \) as:
- For \( x \geq 2 \):
\[ f(x) = 1 - (x - 2) = 3 - x \]
- For \( x < 2 \):
\[ f(x) = 1 - (2 - x) = x - 1 \]

### Step 3: Analyze Each Piece
Now we will analyze the two pieces of the function separately to determine their ranges.

1. **For \( x \geq 2 \)**:
\[ f(x) = 3 - x \]
As \( x \) increases from \( 2 \) to \( \infty \), \( f(x) \) decreases from \( 1 \) to \( -\infty \). Therefore, the range for this piece is:
\[ (-\infty, 1] \]

2. **For \( x < 2 \)**:
\[ f(x) = x - 1 \]
As \( x \) decreases from \( 2 \) to \( -\infty \), \( f(x) \) increases from \( 1 \) to \( -\infty \). Therefore, the range for this piece is:
\[ (-\infty, 1) \]

### Step 4: Combine the Ranges
The overall range of the function \( f(x) \) combines the ranges from both pieces:
- From the first piece, we have \( (-\infty, 1] \)
- From the second piece, we have \( (-\infty, 1) \)

Since both ranges extend to \( -\infty \) and the maximum value is \( 1 \), the combined range of \( f(x) \) is:
\[ (-\infty, 1] \]

### Final Answer
The range of the function \( f(x) = 1 - |x - 2| \) is:
\[ (-\infty, 1] \]