The given function is:

$f(x) = \frac{x - 3}{2x - 5}$

To find what number must be excluded from the domain of $f$, we need to determine when the denominator becomes zero, as division by zero is undefined in mathematics.

The denominator is $2x - 5$. Set this equal to zero and solve for $x$:

$2x - 5 = 0$

$2x = 5$

$x = \frac{5}{2}$

So, $x = \frac{5}{2}$ must be excluded from the domain because it would make the denominator zero, which is undefined.

Answer:

The number that must be excluded from the domain is $x = \frac{5}{2}$.

Would you like any further details on this or have any other questions?

Here are 5 related questions you might explore:

1. What is the domain of $f(x)$ excluding the value $x = \frac{5}{2}$?
2. How does the behavior of the function change near $x = \frac{5}{2}$?
3. Can you express the function's behavior as $x \to \frac{5}{2}$ from the left and right?
4. What happens to $f(x)$ when $x = 0$?
5. How can you graph this rational function and mark its vertical asymptote?

Tip: When working with rational functions, always check for values that make the denominator zero, as these values need to be excluded from the domain.