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: Find the Domain
The domain of a function is the set of all possible input values (x-values) for which the function is defined. Since this is a rational function, we need to ensure that the denominator is not equal to zero.

1. Identify the denominator: \( 2 - x \).
2. Set the denominator not equal to zero:
\[ 2 - x \neq 0 \]
3. Solve for \( x \):
\[ x \neq 2 \]
4. Therefore, the domain of the function is all real numbers except \( x = 2 \):
\[ \text{Domain} = \mathbb{R} - \{2\} \]

### Step 2: Find the Range
To find the range, we will express \( y \) in terms of \( x \) and then determine which values \( y \) can take.

1. Start with the equation:
\[ y = \frac{x-3}{2-x} \]
2. Cross-multiply to eliminate the fraction:
\[ y(2 - x) = x - 3 \]
3. Expand and rearrange the equation:
\[ 2y - xy = x - 3 \]
\[ 2y + 3 = x + xy \]
4. Factor out \( x \) from the right side:
\[ 2y + 3 = x(1 + y) \]
5. Solve for \( x \):
\[ x = \frac{2y + 3}{1 + y} \]
6. Now, we need to determine for which values of \( y \) this expression is valid. The denominator \( 1 + y \) must not be zero:
\[ 1 + y \neq 0 \implies y \neq -1 \]
7. Therefore, the range of the function is all real numbers except \( y = -1 \):
\[ \text{Range} = \mathbb{R} - \{-1\} \]

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