We are given the function:

$f(x) = \frac{x(x-2)}{x-1}$

and are tasked to find:

$f(0) + f(-1) + f(-4) + f(2) - \frac{5}{2}$

Let's evaluate the function at each value of $x$ separately:

1. $f(0)$

$f(0) = \frac{0(0-2)}{0-1} = \frac{0}{-1} = 0$

2. $f(-1)$

$f(-1) = \frac{-1(-1-2)}{-1-1} = \frac{-1(-3)}{-2} = \frac{3}{-2} = -\frac{3}{2}$

3. $f(-4)$

$f(-4) = \frac{-4(-4-2)}{-4-1} = \frac{-4(-6)}{-5} = \frac{24}{-5} = -\frac{24}{5}$

4. $f(2)$

Here, $f(2)$ is undefined because the denominator $x - 1$ becomes zero when $x = 2$. Therefore, $f(2)$ is not defined.

Putting it all together:

Since $f(2)$ is undefined, the entire expression becomes undefined.

Thus, the final result is undefined due to $f(2)$.

Would you like more details or have any questions about this?

Here are 5 related questions:

1. What is the behavior of the function $f(x)$ as $x \to 1$?
2. How would you handle limits in this case?
3. Can the discontinuity at $x = 1$ be removed by redefining $f(x)$?
4. How would you compute the limit of $f(x)$ as $x \to 2$?
5. What is the domain of the function $f(x)$?

Tip: Always check the domain of a function before evaluating it at specific points to avoid undefined values.