The domain and range of the real function f defined by `f(x)=(x-2)/(2-x)` are

Options

A. Domain = R - {2}, Range = {-1}
B. Domain = R-{-2), Range = {-1}
C. Domain = R{2}, Range = {1}
D. Domain =R- {2}, Range = {1}

Answer

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

### Step 1: Identify the function
We start with the function:
\[ f(x) = \frac{x-2}{2-x} \]

### Step 2: Determine the domain
The domain of a function consists of all the values of \( x \) for which the function is defined. The function \( f(x) \) is undefined when the denominator is zero.

Set the denominator equal to zero:
\[ 2 - x = 0 \]
Solving for \( x \):
\[ x = 2 \]
Thus, the function is undefined at \( x = 2 \). Therefore, the domain of \( f(x) \) is all real numbers except 2:
\[ \text{Domain} = \mathbb{R} - \{2\} \]

### Step 3: Simplify the function
Next, we can simplify the function:
\[ f(x) = \frac{x-2}{2-x} = \frac{x-2}{-(x-2)} = -1 \quad \text{for } x \neq 2 \]
This shows that \( f(x) \) is equal to \(-1\) for all \( x \) in the domain.

### Step 4: Determine the range
Since \( f(x) = -1 \) for all \( x \) in the domain, the range of the function is simply:
\[ \text{Range} = \{-1\} \]

### Conclusion
Thus, we have determined that:
- The domain of \( f(x) \) is \( \mathbb{R} - \{2\} \)
- The range of \( f(x) \) is \{-1\}

### Final Answer
The domain and range of the function \( f(x) = \frac{x-2}{2-x} \) are:
- Domain: \( \mathbb{R} - \{2\} \)
- Range: \{-1\}

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