Domain and range of `f (x) = (|x-3|)/(x-3)` are respectively

Options

A. `R, [-1,1]`
B. `R- {3}, {1,-1}`
C. `R ^(+),R`
D. None of these

Answer

To find the domain and range of the function \( f(x) = \frac{|x-3|}{x-3} \), 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.

1. The function \( f(x) = \frac{|x-3|}{x-3} \) is defined for all \( x \) except where the denominator is zero.
2. Set the denominator equal to zero to find the restriction:
\[ x - 3 = 0 \implies x = 3 \]
3. Therefore, the function is undefined at \( x = 3 \).

Thus, the domain of \( f(x) \) is:
\[ \text{Domain} = \{ x \in \mathbb{R} : x \neq 3 \} \]

### Step 2: Determine the Range
To find the range, we analyze the behavior of the function based on the value of \( x \).

1. **Case 1: When \( x \geq 3 \)**
- In this case, \( |x-3| = x-3 \).
- Therefore,
\[ f(x) = \frac{x-3}{x-3} = 1 \quad \text{(for } x > 3\text{)} \]
- At \( x = 3 \), the function is undefined.

2. **Case 2: When \( x < 3 \)**
- Here, \( |x-3| = -(x-3) = 3-x \).
- Thus,
\[ f(x) = \frac{3-x}{x-3} = \frac{-(x-3)}{x-3} = -1 \quad \text{(for } x < 3\text{)} \]

From the analysis:
- For \( x \geq 3 \), \( f(x) = 1 \).
- For \( x < 3 \), \( f(x) = -1 \).

Thus, the range of \( f(x) \) is:
\[ \text{Range} = \{ -1, 1 \} \]

### Final Answer
- **Domain**: \( \{ x \in \mathbb{R} : x \neq 3 \} \)
- **Range**: \( \{ -1, 1 \} \)

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