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Rational functions are an important class of functions in mathematics. They are defined as the ratio between two polynomials. In simpler terms, a rational function looks something like this: \( f(x) = \frac{P(x)}{Q(x)} \) where both \( P(x) \) and \( Q(x) \) are polynomials. These functions are frequently used because they can model relationships in various fields, such as economics or engineering.
One key feature of rational functions is that they are only defined when the denominator is not equal to zero. This is because a division by zero is undefined in mathematics. Therefore, the values of \( x \) which make \( Q(x) = 0 \) are excluded from the domain of the function.

Vertical asymptotes are lines that the graph of a function approaches but never actually touches or crosses as it heads towards infinity. For rational functions specifically, vertical asymptotes occur at values where the denominator is equal to zero but the numerator is not zero. These values make the function undefined and often cause the graph to spike upwards or downwards indefinitely.
To find these vertical asymptotes, you resolve the equation \( Q(x) = 0 \). For the function \( f(x) = \frac{2x}{x^2-x-6} \), the denominator \( x^2-x-6 \) factors into \((x-3)(x+2)\). Hence, the vertical asymptotes are at \( x = 3 \) and \( x = -2 \), because those are the points where the denominator becomes zero.

Horizontal asymptotes describe the behavior of a graph as \( x \) approaches infinity. Unlike vertical asymptotes, the graph of a function can intersect horizontal asymptotes in its finite range but will eventually approach them infinitely. In rational functions, horizontal asymptotes depend on the degrees of the numerator and the denominator.
There are three key cases to consider:

• 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 is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients.
• If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote, but there might be an oblique asymptote instead.

For \( f(x) = \frac{2x}{x^2-x-6} \), the degree of the numerator is \( 1 \) and the degree of the denominator is \( 2 \). Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y=0 \).

Understanding polynomials is crucial for mastering rational functions. Polynomials are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. For example, \( x^2 - x - 6 \) is a polynomial with a degree of 2, since the highest power of \( x \) is 2. Each individual term in a polynomial is called a monomial, and they can include constants, variables, and exponents.
Polynomials can be classified by their degree, which is the highest exponent of the variable. The degree plays a significant role in determining the behavior of rational functions, including vertical and horizontal asymptotes. Moreover, when dealing with rational functions, working with polynomials involves performing operations such as addition, subtraction, multiplication, and factoring. Factoring is particularly useful for finding the roots of a polynomial, which are the values of \( x \) that make the polynomial equal to zero.