Find the domain and range of the following functions:
`(x^(2)-4)/(x-2)`

Answer

To find the domain and range of the function \( f(x) = \frac{x^2 - 4}{x - 2} \), we will follow these steps:

### Step 1: Identify the Domain
1. The function is a rational function, which means it is defined everywhere except where the denominator is zero.
2. Set the denominator equal to zero to find the point of discontinuity:
\[ x - 2 = 0 \implies x = 2 \]
3. Therefore, the function is not defined at \( x = 2 \).
4. The domain of the function is all real numbers except \( x = 2 \). In interval notation, this can be expressed as:
\[ \text{Domain} = (-\infty, 2) \cup (2, \infty) \]

### Step 2: Simplify the Function
1. Factor the numerator:
\[ x^2 - 4 = (x - 2)(x + 2) \]
2. Substitute this back into the function:
\[ f(x) = \frac{(x - 2)(x + 2)}{x - 2} \]
3. For \( x \neq 2 \), we can cancel \( (x - 2) \) in the numerator and denominator:
\[ f(x) = x + 2 \quad \text{for } x \neq 2 \]

### Step 3: Identify the Range
1. The simplified function \( f(x) = x + 2 \) is a linear function, which is defined for all \( x \) in its domain.
2. The output of the function will be all real numbers except for the value when \( x = 2 \):
\[ f(2) = 2 + 2 = 4 \]
3. Thus, the range of the function excludes the value \( 4 \):
\[ \text{Range} = (-\infty, 4) \cup (4, \infty) \]

### Final Answer
- **Domain**: \( (-\infty, 2) \cup (2, \infty) \)
- **Range**: \( (-\infty, 4) \cup (4, \infty) \)