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Rational functions are a type of mathematical function represented as a fraction. They can be expressed as \( r(x) = \frac{p(x)}{q(x)} \), where \( p(x) \) and \( q(x) \) are polynomials. The key aspect of rational functions is that the denominator, \( q(x) \), must not be zero for any real number, as division by zero is undefined.

• Numerator \( p(x) \): Determines the behavior of the function and plays a role in simplifying the expression.
• Denominator \( q(x) \): Influences the domain and potential asymptotes.

It's helpful to understand that "fractures" in the function, where \( q(x) = 0 \), will either result in vertical asymptotes or holes, depending on factorization. In the example \( r(x) = \frac{6x}{x^2 + 2} \), the denominator has no real roots (since \( x^2 + 2 = 0 \) leads to \( x^2 = -2 \)), meaning there are no vertical asymptotes. This example illustrates a simple case where the degrees of \( p(x) \) and \( q(x) \) greatly influence the type of asymptote found.

Horizontal asymptotes are lines parallel to the x-axis that a graph approaches but never quite reaches. These occur when evaluating the end behavior of a rational function as the input \( x \) approaches positive or negative infinity. The general rules for finding horizontal asymptotes depend on the degrees of the numerator and the denominator:

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \).
• If the degree of the numerator equals the degree of the denominator, the horizontal asymptote equals the leading coefficients' ratio.
• If the degree of the numerator is greater, there's no horizontal asymptote.

For the function \( r(x) = \frac{6x}{x^2 + 2} \), the denominator's degree (2) is greater than the numerator's degree (1). Thus, its horizontal asymptote is \( y = 0 \), meaning as \( x \to \pm \infty \), \( r(x) \) approaches zero.

Vertical asymptotes are vertical lines that the graph of a function approaches but does not touch. They occur where the denominator of a rational function equals zero, provided that the numerator does not also equal zero at that point. This causes the function's value to increase or decrease without bound, or in simpler terms, to "shoot off" towards infinity or negative infinity. To locate vertical asymptotes, solve \( q(x) = 0 \) for \( x \), where \( q(x) \) is the denominator. However, the example \( r(x) = \frac{6x}{x^2 + 2} \) presents a unique situation where \( x^2 + 2 = 0 \) gives \( x^2 = -2 \), a condition that has no real solution. As a result, \( r(x) \) has no vertical asymptotes. This demonstrates an important aspect: not all rational functions have vertical asymptotes, particularly when their denominators have no real roots.