To find the range of the function $f(x) = \frac{3x - 5}{x - 2}$, let's analyze its behavior and identify any restrictions on the values it can take.

1. Identify Domain Restrictions: The function is undefined when $x = 2$ because the denominator $x - 2 = 0$ at this point. Therefore, $x = 2$ is not in the domain of $f(x)$.

2. Rewrite the Function to Understand Behavior: $f(x) = \frac{3x - 5}{x - 2}$ To simplify, we can perform polynomial division: $\frac{3x - 5}{x - 2} = 3 + \frac{1}{x - 2}$ This tells us that $f(x) = 3 + \frac{1}{x - 2}$.

3. Analyze the Range:

   • As $x$ approaches 2 from either side, the term $\frac{1}{x - 2}$ will approach $\pm \infty$, so $f(x)$ will approach $\pm \infty$ as well.
   • For other values of $x$, $\frac{1}{x - 2}$ can take any real value except zero (since $x \neq 2$), and so $f(x) = 3 + \frac{1}{x - 2}$ can take any real value.

4. Conclusion on the Range: Since $f(x)$ can approach any real number as $x$ varies (except at the undefined point $x = 2$), the range of $f(x)$ is all real numbers: $\text{Range of } f(x) = (-\infty, \infty)$

Would you like more details or have any questions?

Here are 5 related questions that expand on this concept:

1. How do we find the range of a more complex rational function?
2. What are the steps to identify horizontal and vertical asymptotes in rational functions?
3. How does polynomial division help simplify rational functions?
4. What is the domain of a rational function, and how do we determine it?
5. How does analyzing limits help in determining the range of a function?

Tip: To find the range of a rational function, consider simplifying it and analyzing any asymptotic behavior, as this often reveals the range directly.