Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. ƒ(x)=(x²-1)/(x+3)

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Identify the rational function given: \(f(x) = \frac{x^2 - 1}{x + 3}\).

Find the vertical asymptotes by setting the denominator equal to zero and solving for \(x\): solve \(x + 3 = 0\) to find values where the function is undefined.

Determine the horizontal or oblique asymptotes by comparing the degrees of the numerator and denominator polynomials. Here, the numerator degree is 2 and the denominator degree is 1.

Since the numerator degree is greater than the denominator degree by exactly 1, perform polynomial long division of \(x^2 - 1\) by \(x + 3\) to find the oblique asymptote.

Write the oblique asymptote as the quotient from the division (ignoring the remainder), and summarize the vertical asymptote(s) found in step 2.

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주요 개념

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Vertical Asymptotes

Vertical asymptotes occur where the denominator of a rational function equals zero, causing the function to approach infinity or negative infinity. To find them, set the denominator equal to zero and solve for x, ensuring these values do not cancel with factors in the numerator.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as x approaches infinity or negative infinity. For rational functions, compare the degrees of the numerator and denominator: if the numerator's degree is less, the asymptote is y=0; if equal, it's the ratio of leading coefficients.

Oblique (Slant) Asymptotes

Oblique asymptotes occur when the degree of the numerator is exactly one more than the denominator. They are found by performing polynomial division of numerator by denominator; the quotient (without the remainder) gives the equation of the slant asymptote.

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. ƒ(x)=(x2-1)/(x+3)