Find the domain of the range of each of the following real functions:
`f(x)=1-|x-2|`

Explanation

To find the domain and range of the function \( f(x) = 1 - |x - 2| \), we will follow these steps:

### Step 1: Understand the function
The function involves an absolute value, which can be split into two cases based on the value of \( x \).

### Step 2: Split the function into cases
1. **Case 1:** When \( x \geq 2 \)
\[ f(x) = 1 - (x - 2) = 1 - x + 2 = 3 - x \]

2. **Case 2:** When \( x < 2 \)
\[ f(x) = 1 - (-(x - 2)) = 1 + x - 2 = x - 1 \]

### Step 3: Determine the domain
The domain of \( f(x) \) is the set of all real numbers for which the function is defined. Since there are no restrictions on \( x \) in either case, the domain is:
\[ \text{Domain} = (-\infty, \infty) \]

### Step 4: Determine the range
Now we will find the range of the function by analyzing both cases:

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 output values in this case are:
\[ f(x) \in (-\infty, 1] \]

2. **For \( x < 2 \):**
\[ f(x) = x - 1 \]
As \( x \) approaches \( 2 \) from the left, \( f(x) \) approaches \( 1 \). As \( x \) decreases towards \( -\infty \), \( f(x) \) also decreases towards \( -\infty \). Therefore, the output values in this case are:
\[ f(x) \in (-\infty, 1) \]

### Step 5: Combine the ranges
Since both cases yield the same upper limit of \( 1 \) and extend down to \( -\infty \), the overall range of the function is:
\[ \text{Range} = (-\infty, 1] \]

### Final Answer:
- **Domain:** \( (-\infty, \infty) \)
- **Range:** \( (-\infty, 1] \)