Find the domain and range of the following functions:
`(x-3)/(2-x)`

Answer

To find the domain and range of the function \( f(x) = \frac{x-3}{2-x} \), we will follow these steps:

### Step 1: Determine the Domain
The domain of a function consists of all the values of \( x \) for which the function is defined. In this case, we need to ensure that the denominator does not equal zero.

1. Set the denominator equal to zero:
\[ 2 - x = 0 \]
Solving for \( x \):
\[ x = 2 \]

2. Since the function is undefined at \( x = 2 \), the domain of \( f(x) \) is all real numbers except 2. This can be expressed in set notation:
\[ \text{Domain} = \mathbb{R} - \{2\} \]

### Step 2: Determine the Range
To find the range of the function, we will express \( x \) in terms of \( y \) and analyze the resulting expression.

1. Start with the equation:
\[ y = \frac{x-3}{2-x} \]

2. Rearranging the equation to solve for \( x \):
\[ y(2 - x) = x - 3 \]
Expanding this gives:
\[ 2y - xy = x - 3 \]

3. Rearranging terms to isolate \( x \):
\[ xy + x = 2y + 3 \]
Factor out \( x \):
\[ x(y + 1) = 2y + 3 \]

4. Solving for \( x \):
\[ x = \frac{2y + 3}{y + 1} \]

5. The expression \( y + 1 \) cannot be zero (as it is in the denominator), so we set:
\[ y + 1 \neq 0 \implies y \neq -1 \]

6. Therefore, the range of \( f(x) \) is all real numbers except -1:
\[ \text{Range} = \mathbb{R} - \{-1\} \]

### Final Answer
- **Domain**: \( \mathbb{R} - \{2\} \)
- **Range**: \( \mathbb{R} - \{-1\} \)