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 values of \( x \) for which the function is defined.

1. **Identify the denominator**: The function is undefined when the denominator is zero.
\[ x - 3 = 0 \implies x = 3 \]
Therefore, the function is undefined at \( x = 3 \).

2. **Write the domain**: Since the function is defined for all real numbers except \( x = 3 \), the domain can be expressed as:
\[ \text{Domain} = \mathbb{R} - \{3\} \]

### Step 2: Simplify the Function
Next, we simplify the function to find its range.

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**: Substitute the factored form into the function:
\[ f(x) = \frac{(x - 3)(x + 3)}{x - 3} \]
For \( x \neq 3 \), we can simplify this to:
\[ f(x) = x + 3 \]

### Step 3: Identify the Range
Now, we will find the range of the simplified function \( f(x) = x + 3 \).

1. **Determine the output values**: The function \( f(x) = x + 3 \) is a linear function with a slope of 1. It can take any real number value as \( x \) varies over its domain.

2. **Exclusion of a specific value**: Since \( f(3) = 6 \) is not defined (as \( x = 3 \) is excluded from the domain), the value \( 6 \) is not included in the range.

3. **Write the range**: Thus, the range of the function can be expressed as:
\[ \text{Range} = \mathbb{R} - \{6\} \]

### Final Answer:
- **Domain**: \( \mathbb{R} - \{3\} \)
- **Range**: \( \mathbb{R} - \{6\} \)