To determine the vertical asymptotes of the function $f(x) = \frac{3x - 5}{x - 2}$, we need to examine the points at which the function becomes undefined. Vertical asymptotes occur where the denominator of the function is zero, as long as the numerator does not also become zero at that same point (this would instead indicate a hole in the graph).

1. Identify when the denominator is zero: Set the denominator equal to zero: $x - 2 = 0$ Solving this gives: $x = 2$

2. Check the numerator at $x = 2$: Substitute $x = 2$ into the numerator: $3(2) - 5 = 6 - 5 = 1 \neq 0$ Since the numerator does not equal zero when $x = 2$, there is indeed a vertical asymptote at $x = 2$.

Since this is the only value of $x$ that makes the denominator zero, we conclude that the function has exactly one vertical asymptote at $x = 2$.

Answer

The correct choice is: B. The function has one vertical asymptote, $x = 2$.

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

Here are five related questions you may find helpful:

1. How do you determine horizontal asymptotes for rational functions?
2. What is the behavior of $f(x)$ near its vertical asymptote at $x = 2$?
3. How can you identify holes in the graph of a rational function?
4. How do you find the slant (oblique) asymptote of a rational function if it exists?
5. What would change in the function if the numerator and denominator both had a common factor?

Tip: For rational functions, always check if the numerator and denominator share common factors before concluding the presence of vertical asymptotes or holes.