In Exercises 71–78, find the vertical asymptotes, if any,...

In Exercises 71–78, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function.
$$
g(x)=\frac{2 x-4}{x+3}
$$

Key Concepts

Rational Functions

A rational function is a quotient of two polynomials where the domain consists of all real numbers except those that make the denominator zero. Understanding its structure is fundamental for analyzing its behavior and key features such as asymptotes, intercepts, and discontinuities.

Vertical Asymptotes

Vertical asymptotes occur at values of x that make the denominator zero, provided these values are not canceled by the numerator. They represent lines where the function's value increases or decreases without bound, and indicate points of discontinuity in the function.

Horizontal Asymptotes

Horizontal asymptotes describe the end behavior of a rational function as x approaches positive or negative infinity. They are determined by comparing the degrees of the numerator and the denominator: if the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients; if the numerator’s degree is lower, the asymptote is y = 0; and if greater, a horizontal asymptote does not exist.

Slant (Oblique) Asymptotes

A slant or oblique asymptote arises when the degree of the numerator is exactly one more than the degree of the denominator. It is found through polynomial long division, where the quotient (ignoring the remainder) gives the equation of the asymptote, which describes the linear trend of the function as x approaches infinity or negative infinity.

Graphing Rational Functions

Graphing a rational function involves using identified features such as vertical, horizontal, and slant asymptotes along with intercepts and behavior near the asymptotes to sketch the overall behavior of the function. This comprehensive approach helps in visualizing how the function behaves across its entire domain.

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