1. Let's start by defining the **asymptote**. An asymptote is a line that a graph approaches but never actually touches or crosses as the input or output values become very large or very small. 2. There are three main types of asymptotes: - **Vertical asymptotes** occur where the function grows without bound as the input approaches a certain value. - **Horizontal asymptotes** describe the value the function approaches as the input goes to positive or negative infinity. - **Oblique (slant) asymptotes** happen when the function approaches a line that is neither vertical nor horizontal. 3. Next, the **domain** of a function is the set of all possible input values (usually x-values) for which the function is defined. 4. The **range** of a function is the set of all possible output values (usually y-values) that the function can produce. 5. To find the domain, look for values that make the function undefined, such as division by zero or taking the square root of a negative number. 6. To find the range, analyze the behavior of the function and its outputs, often using the graph or algebraic manipulation. 7. Understanding asymptotes helps in determining the behavior of the function at extreme values, which also influences the range. In summary: - **Asymptotes** describe the behavior of the function near certain lines. - **Domain** is all valid inputs. - **Range** is all possible outputs.