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What is a rational function?

Back

A rational function is a function that can be expressed as the quotient of two polynomials, i.e., \( f(x) = \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are polynomials.

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What are vertical asymptotes?

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Vertical asymptotes are lines \( x = a \) where a rational function approaches infinity or negative infinity as the input approaches \( a \). They occur where the denominator is zero and the numerator is not zero.

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How do you find vertical asymptotes of a rational function?

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To find vertical asymptotes, set the denominator equal to zero and solve for \( x \). The values of \( x \) that make the denominator zero are the vertical asymptotes.

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What are horizontal asymptotes?

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Horizontal asymptotes are lines \( y = b \) that a rational function approaches as \( x \) approaches infinity or negative infinity. They describe the end behavior of the function.

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How do you find horizontal asymptotes of a rational function?

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To find horizontal asymptotes, compare the degrees of the numerator and denominator: 1) If the degree of the numerator is less than the degree of the denominator, \( y = 0 \). 2) If they are equal, \( y = \frac{a}{b} \) where \( a \) and \( b \) are the leading coefficients. 3) If the degree of the numerator is greater, there is no horizontal asymptote.

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What is the significance of the x-intercept in a rational function?

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The x-intercept is the point where the graph of the function crosses the x-axis, found by setting the numerator equal to zero and solving for \( x \).

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What is the significance of the y-intercept in a rational function?

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The y-intercept is the point where the graph of the function crosses the y-axis, found by evaluating the function at \( x = 0 \).

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How do you determine the end behavior of a rational function?

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The end behavior can be determined by analyzing the degrees of the numerator and denominator and their leading coefficients, which dictate the horizontal asymptote.

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What is a hole in a rational function?

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A hole occurs in a rational function at a value of \( x \) that makes both the numerator and denominator zero. It indicates a removable discontinuity.

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What is the domain of a rational function?

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The domain of a rational function is all real numbers except for the values that make the denominator zero.