Find the range fo the function `f(x)=(x^(2)-x-6)/(x-3)`

Explanation

To find the range of the function \( f(x) = \frac{x^2 - x - 6}{x - 3} \), we can follow these steps:

### Step 1: Factor the numerator
First, we need to factor the numerator \( x^2 - x - 6 \). We can rewrite it as:
\[ x^2 - x - 6 = (x - 3)(x + 2) \]
Thus, we can rewrite the function as:
\[ f(x) = \frac{(x - 3)(x + 2)}{x - 3} \]

### Step 2: Simplify the function
Since \( x \neq 3 \) (to avoid division by zero), we can cancel \( (x - 3) \) from the numerator and denominator:
\[ f(x) = x + 2 \quad \text{for } x \neq 3 \]

### Step 3: Identify the type of function
The function \( f(x) = x + 2 \) is a linear function, which means it can take any real value as output except for the value at \( x = 3 \).

### Step 4: Find the value of the function at the excluded point
We need to find the value of \( f(x) \) when \( x = 3 \):
\[ f(3) = 3 + 2 = 5 \]
However, since \( x = 3 \) is not in the domain of the function, \( f(3) \) is not included in the range.

### Step 5: Determine the range
Since \( f(x) = x + 2 \) can take any real value except for 5, the range of the function is:
\[ \text{Range} = \mathbb{R} - \{5\} \]

### Final Answer
Thus, the range of the function \( f(x) \) is:
\[ \text{Range} = (-\infty, 5) \cup (5, \infty) \]

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