Asymptotes are important when figuring out the domain and range of certain functions, especially rational and some special types called transcendental functions. So, what are asymptotes? They are lines that a function gets close to but never actually meets. There are three main types of asymptotes: vertical, horizontal, and oblique (or slant).

1. Vertical Asymptotes (VA)
Vertical asymptotes show where the domain (the input values) cannot go. For example, in the function $f(x) = \frac{1}{x-3}$, the vertical asymptote is at $x = 3$. This means that the number 3 is not included in the domain. We can write this like this:
$\text{Domain: } (-\infty, 3) \cup (3, \infty)$

2. Horizontal Asymptotes (HA)
Horizontal asymptotes help us find the range (the output values) of the function when $x$ becomes very large or very small. For the same function, $f(x) = \frac{1}{x-3}$, the horizontal asymptote is at $y = 0$. This tells us that as $x$ goes to positive or negative infinity, the function gets close to zero but never really reaches it. Therefore, we can say the range is:
$\text{Range: } (-\infty, 0) \cup (0, \infty)$

3. Oblique Asymptotes (OA)
Oblique asymptotes happen when the top part (numerator) of the function has a higher degree than the bottom part (denominator) by one. They also give us useful information about what happens to the function as $x$ goes to infinity.

To wrap it all up, when looking at asymptotes, keep in mind:

• Vertical asymptotes show limits on the domain.
• Horizontal and oblique asymptotes tell us about the range.

Knowing these ideas helps us draw functions correctly and understand how they behave!