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The domain of a rational function refers to all the possible values of the variable, typically denoted as \( x \), for which the function is defined. In simpler terms, it represents all the \( x \) values you can plug into the function without causing any mathematical errors.
For the rational function \( f(x) = \frac{4}{x-1} \), we need to focus on the denominator, \( x-1 \), because division by zero is undefined. To find where the function is undefined, solve \( x-1=0 \), which gives \( x=1 \).
This indicates that \( x=1 \) is not included in the domain of \( f(x) \). Therefore, the domain is all real numbers excluding \( x=1 \), expressed as \( x \in \mathbb{R}, \ x eq 1 \).

• Key Point: Domains exclude values where the denominator equals zero.
• Important: Always solve for \( x \) in the denominator to define the domain correctly.

Vertical asymptotes occur at the values of \( x \) that make the denominator of a rational function equal to zero. These are points where the function heads towards positive or negative infinity, indicating a sharp curve on the graph.
In the function \( f(x) = \frac{4}{x-1} \), we've determined that the denominator becomes zero at \( x=1 \). Consequently, a vertical asymptote exists at this point.
A vertical asymptote essentially creates a boundary line on the graph that the function approaches but never crosses.

• Vertical asymptotes tell us where the function spikes abruptly.
• These asymptotes occur at values excluded from the domain.

Horizontal asymptotes provide insight into the behavior of a function as \( x \) becomes extremely large, either positively or negatively. To determine their presence in a rational function, compare the degrees of the numerator and the denominator.
For \( f(x) = \frac{4}{x-1} \), the numerator is a constant, indicating a degree of zero, while the denominator has a degree of one. When the degree of the denominator is higher than the numerator, \( f(x) \) approaches zero as \( x \to \pm \infty \). This means the horizontal asymptote exists at \( y=0 \).
Horizontal asymptotes guide us on how the function levels off at its tails and provide a sense of stability as \( x \) increases or decreases without bound.

• If the numerator's degree is less than the denominator's, expect a horizontal asymptote at \( y=0 \).
• These asymptotes help predict long-term behavior of rational functions.