Find the domain and range of `f(x)=|x-1|`

Answer

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

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

For the function \( f(x) = |x - 1| \):
- The absolute value function is defined for all real numbers.
- Therefore, there are no restrictions on the values of \( x \).

Thus, the domain of \( f(x) \) is:
\[ \text{Domain} = (-\infty, \infty) \]

### Step 2: Determine the Range
The range of a function is the set of all possible output values (f(x)-values).

For the function \( f(x) = |x - 1| \):
- The absolute value function always produces non-negative outputs.
- The minimum value of \( f(x) \) occurs when \( x = 1 \), where \( f(1) = |1 - 1| = 0 \).
- As \( x \) moves away from 1 in either direction, \( f(x) \) increases without bound.

Thus, the range of \( f(x) \) is:
\[ \text{Range} = [0, \infty) \]

### Final Answer
- **Domain**: \( (-\infty, \infty) \)
- **Range**: \( [0, \infty) \)