1. Let's start by defining the **domain** of a function. The domain is the set of all possible input values (usually $x$) for which the function is defined. 2. For example, if you have a function $f(x) = \frac{1}{x}$, the domain is all real numbers except $x=0$ because division by zero is undefined. 3. The **range** of a function is the set of all possible output values (usually $y$) that the function can produce. 4. Using the same function $f(x) = \frac{1}{x}$, the range is all real numbers except $y=0$ because $\frac{1}{x}$ never equals zero. 5. Now, an **asymptote** is a line that the graph of a function approaches but never touches or crosses as $x$ or $y$ goes to infinity or some value. 6. There are three types of asymptotes: vertical, horizontal, and oblique (slant). 7. A **vertical asymptote** occurs where the function is undefined and the values of $f(x)$ grow without bound, like $x=0$ for $f(x) = \frac{1}{x}$. 8. A **horizontal asymptote** describes the behavior of $f(x)$ as $x$ approaches infinity or negative infinity, such as $y=0$ for $f(x) = \frac{1}{x}$. 9. To summarize: - Domain: all valid $x$ values. - Range: all possible $y$ values. - Asymptote: a line the graph approaches but does not cross. Understanding these concepts helps analyze and graph functions effectively.