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
The first step is to factor the numerator \( x^2 - x - 6 \). We can rewrite it as:
\[ x^2 - x - 6 = (x - 3)(x + 2) \]

### Step 2: Rewrite the function
Now, we can substitute the factored form back into the function:
\[ f(x) = \frac{(x - 3)(x + 2)}{x - 3} \]

### Step 3: Simplify the function
For \( x \neq 3 \), we can simplify the function:
\[ f(x) = x + 2 \]
However, we note that \( f(x) \) is not defined at \( x = 3 \) because it leads to a \( \frac{0}{0} \) form.

### Step 4: Determine the range of the simplified function
The function \( f(x) = x + 2 \) is a linear function, which means it can take any real number value. The range of \( x + 2 \) is all real numbers \( \mathbb{R} \).

### Step 5: Exclude the value at the undefined point
Since \( f(x) \) is not defined at \( x = 3 \), we need to find the value of \( f(3) \):
\[ f(3) = 3 + 2 = 5 \]
Thus, the value 5 is not in the range of the function.

### Conclusion: State the range
The range of the function \( f(x) \) is all real numbers except 5. We can express this as:
\[ \text{Range of } f(x) = \mathbb{R} \setminus \{5\} \]