Find the domain and range of the function `f(x)=(x^(2)-9)/(x-3)`

Explanation

To find the domain and range of the function \( f(x) = \frac{x^2 - 9}{x - 3} \), we will follow these steps:

### Step 1: Identify the Domain
The domain of a function consists of all the possible values of \( x \) for which the function is defined.

1. **Check for restrictions**: The function \( f(x) \) is a rational function, which means it can be undefined if the denominator is zero.
- Set the denominator equal to zero:
\[ x - 3 = 0 \implies x = 3 \]
- Therefore, the function is undefined at \( x = 3 \).

2. **Write the domain**: The domain is all real numbers except \( x = 3 \).
\[ \text{Domain: } x \in \mathbb{R}, x \neq 3 \]

### Step 2: Simplify the Function
Next, we can simplify the function to better understand its behavior.

1. **Factor the numerator**: The numerator \( x^2 - 9 \) can be factored using the difference of squares:
\[ x^2 - 9 = (x - 3)(x + 3) \]

2. **Rewrite the function**:
\[ f(x) = \frac{(x - 3)(x + 3)}{x - 3} \]
For \( x \neq 3 \), we can cancel \( x - 3 \):
\[ f(x) = x + 3 \quad \text{for } x \neq 3 \]

### Step 3: Identify the Range
Now that we have simplified the function, we can find the range.

1. **Determine the output of the simplified function**:
- The simplified function \( f(x) = x + 3 \) is defined for all \( x \) except \( x = 3 \).
- If we substitute \( x = 3 \) into the simplified function, we find:
\[ f(3) = 3 + 3 = 6 \]
- However, since \( f(x) \) is not defined at \( x = 3 \), the output \( y = 6 \) is not included in the range.

2. **Write the range**: The range is all real numbers except \( y = 6 \).
\[ \text{Range: } y \in \mathbb{R}, y \neq 6 \]

### Final Answer
- **Domain**: \( x \in \mathbb{R}, x \neq 3 \)
- **Range**: \( y \in \mathbb{R}, y \neq 6 \)