In Exercises 57–64, 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) = (2x - 4)/(x + 3)

검증된 단계별 안내

Step 1: Identify the vertical asymptotes by setting the denominator equal to zero. Solve the equation

x + 3 = 0

to find the x-value(s) where the function is undefined.

Step 2: Determine the horizontal asymptote by comparing the degrees of the numerator and denominator. The numerator

2x-4

has degree 1, and the denominator

x+3

also has degree 1. When the degrees are equal, the horizontal asymptote is given by the ratio of the leading coefficients.

Step 3: Check for a slant (oblique) asymptote. A slant asymptote exists if the degree of the numerator is exactly one more than the degree of the denominator. Since the degrees of the numerator and denominator are the same, there is no slant asymptote in this case.

Step 4: Summarize the asymptotes. The vertical asymptote is at the x-value found in Step 1, the horizontal asymptote is the ratio of the leading coefficients found in Step 2, and there is no slant asymptote as determined in Step 3.

Step 5: To graph the function, plot the asymptotes as dashed lines, and then evaluate the function at several x-values to determine the behavior of the graph near the asymptotes. Use these points to sketch the curve.

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주요 개념

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Vertical Asymptotes

Vertical asymptotes occur in rational functions where the denominator equals zero, leading to undefined values. To find vertical asymptotes, set the denominator of the function to zero and solve for x. These asymptotes indicate values that the function approaches but never reaches, often resulting in a division by zero.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as x approaches infinity or negative infinity. For rational functions, the horizontal asymptote can be determined by comparing the degrees of the numerator and denominator. If the degree of the numerator is less than the denominator, the horizontal asymptote is y=0; if they are equal, it is the ratio of their leading coefficients.

Slant Asymptotes

Slant (or oblique) asymptotes occur when the degree of the numerator is exactly one greater than the degree of the denominator. To find a slant asymptote, perform polynomial long division on the rational function. The quotient (ignoring the remainder) gives the equation of the slant asymptote, which describes the function's behavior as x approaches infinity.