Find the domain and range of real function ,`f(x)=1-|x-3|`

Answer

To find the domain and range of the function \( f(x) = 1 - |x - 3| \), we can follow these steps:

### Step 1: Determine the Domain
The domain of a function consists of all the possible input values (x-values) for which the function is defined.

For the function \( f(x) = 1 - |x - 3| \):
- The absolute value function \( |x - 3| \) is defined for all real numbers \( x \).
- Therefore, \( f(x) \) is also defined for all real numbers.

**Domain:** \( \mathbb{R} \) (all real numbers)

### Step 2: Determine the Range
To find the range, we need to analyze the output values of the function \( f(x) \).

1. **Understanding the Absolute Value**:
- The expression \( |x - 3| \) represents the distance of \( x \) from 3 on the number line. This value is always non-negative, meaning \( |x - 3| \geq 0 \).

2. **Substituting into the Function**:
- Since \( |x - 3| \geq 0 \), we can substitute this into the function:
\[ f(x) = 1 - |x - 3| \]
- This means that \( f(x) \) will take the form:
\[ f(x) \leq 1 \]
- The minimum value of \( |x - 3| \) is 0 (when \( x = 3 \)), which gives:
\[ f(3) = 1 - 0 = 1 \]
- As \( |x - 3| \) increases (as \( x \) moves away from 3), \( f(x) \) decreases. The value of \( |x - 3| \) can increase indefinitely, leading \( f(x) \) to decrease indefinitely.

3. **Finding the Minimum Value**:
- As \( |x - 3| \) approaches infinity, \( f(x) \) approaches negative infinity:
\[ f(x) \to -\infty \text{ as } |x - 3| \to \infty \]

**Range:** \( (-\infty, 1] \)

### Final Answer
- **Domain:** \( \mathbb{R} \) (all real numbers)
- **Range:** \( (-\infty, 1] \)