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Understanding the domain and range of a rational function like \( y = \frac{x-1}{x+5} \) is crucial. The **domain** refers to all possible input values \( x \) that you can substitute into the function without causing any mathematical errors, like division by zero. In our exercise, the denominator \( x+5 \) should not equal zero as this would make the function undefined. Thus, \( x eq -5 \), meaning the domain is all real numbers except \( x = -5 \).For the **range**, consider the output values \( y \) that the function can produce. By examining the behavior as \( x \) approaches restrictions, we observe that \( y \) can indeed be any real number. As \( x \) nears -5 from either direction, \( y \) moves towards either positive or negative infinity. Unlike the domain, there's no boundary keeping \( y \) from assuming any real number value, establishing the range as all real numbers, \( y \in R \).

Graphing the rational function \( y = \frac{x-1}{x+5} \) helps visualize its behavior and understand its characteristics. To graph this function, you can use tools like a graphing calculator or computer software designed for this purpose. By plotting the function, you will notice a unique curve known as a hyperbola.The hyperbola will have distinct branches moving towards infinity. These branches display the behavior near the value that would cause division by zero (in this case, \( x = -5 \)). The graph will clearly show how the function approaches its vertical asymptote without ever crossing it.Remember, when graphing functions like these:

• Identify any intercepts: here, when \( x = 0 \), \( y = -\frac{1}{5} \)
• Plot known values and check the behavior near the asymptotes.
• Ensure to double-check calculations for any critical point where the graph could change direction.

Asymptotes are lines that the graph of a function approaches but never actually touches or crosses. Investigating asymptotes for rational functions like \( y = \frac{x-1}{x+5} \) is fundamental to graphing and understanding their nature. For this function, a **vertical asymptote** occurs where the denominator equals zero. Here, that's \( x = -5 \). Graphically, you represent this as a dashed line and observe that the function will rise or fall sharply, climbing toward positive or negative infinity as it approaches this line.Horizontal asymptotes differ; they help to understand what value \( y \) approaches as \( x \) head towards infinity. For rational functions where the numerator and denominator are polynomials of the same degree, like this example (both linear), the horizontal asymptote can be found by dividing the leading coefficients. Hence, as \( x \to \infty \) or \( x \to -\infty \), \( y \to 1 \) (since the leading coefficient of \( x \) in both the numerator and denominator is 1).These insights provide stability and predictability to the graph, making asymptotes foundational elements for fully understanding and interpreting rational functions.