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 function
We start with the function:
\[ f(x) = \frac{x^2 - 4}{x - 2} \]

### Step 2: Determine the domain
The domain of a rational function is all real numbers except where the denominator is zero.

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

Thus, the domain of \( f(x) \) is all real numbers except \( x = 2 \). In interval notation, this can be expressed as:
\[ \text{Domain} = \mathbb{R} - \{2\} = (-\infty, 2) \cup (2, \infty) \]

### Step 3: Simplify the function
Next, we simplify the function to find the range. Notice that:
\[ x^2 - 4 = (x - 2)(x + 2) \]
Thus, we can rewrite the function as:
\[ f(x) = \frac{(x - 2)(x + 2)}{x - 2} \]
For \( x \neq 2 \), we can cancel \( x - 2 \):
\[ f(x) = x + 2 \quad \text{for } x \neq 2 \]

### Step 4: Determine the range
Now, we need to find the range of the simplified function \( f(x) = x + 2 \).

1. The function \( f(x) = x + 2 \) is a linear function with a slope of 1, which means it can take any real value.
2. However, since \( f(x) \) is undefined at \( x = 2 \), we need to find the value of \( f(2) \):
\[ f(2) = 2 + 2 = 4 \]
Since \( f(x) \) approaches 4 as \( x \) approaches 2, but never actually takes the value 4, we conclude that 4 is excluded from the range.

Thus, the range of \( f(x) \) is:
\[ \text{Range} = \mathbb{R} - \{4\} = (-\infty, 4) \cup (4, \infty) \]

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