Find the domain and range of the function `(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: Simplify the Function
First, we can simplify the function. Notice that \( x^2 - 4 \) can be factored as:
\[ x^2 - 4 = (x - 2)(x + 2) \]
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 2: Determine the Domain
The domain of a function includes all the values of \( x \) for which the function is defined. Since we have cancelled \( x - 2 \), we must exclude \( x = 2 \) from the domain. Therefore, the domain of \( f(x) \) is:
\[ \text{Domain: } (-\infty, 2) \cup (2, \infty) \]

### Step 3: Determine the Range
Next, we analyze the simplified function \( f(x) = x + 2 \). This is a linear function with a slope of 1. As \( x \) approaches \( -\infty \) or \( +\infty \), \( f(x) \) will also approach \( -\infty \) or \( +\infty \) respectively. Therefore, the range of \( f(x) \) is:
\[ \text{Range: } (-\infty, \infty) \]

### Final Result
Thus, we conclude:
- **Domain:** \( (-\infty, 2) \cup (2, \infty) \)
- **Range:** \( (-\infty, \infty) \)

---