Question

if f has a vertical asymptote given by \\(x = 10\\), a horizontal asymptote \\(y = 0\\), y-intercept at \\(-2\\), and no x-intercept, then write the equation of a rational function \\(f(x) = \frac{p(x)}{q(x)}\\) having the indicated properties, in which the degrees of p and q are as small as possible. graph the function using a graphing utility to verify that it has the required properties.

write the equation of the rational function. choose the correct answer below.

a. \\(f(x) = \frac{20}{x + 10\\)}

b. \\(f(x) = \frac{20}{x - 10\\)}

c. \\(f(x) = \frac{10}{x + 10\\)}

d. \\(f(x) = \frac{10}{x - 10\\)}

Explanation:

Analyze vertical asymptote and denominator

Using the Vertical Asymptotes Determination knowledge point

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Analyze horizontal asymptote and degree

Using the Horizontal Asymptotes knowledge point

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Analyze x-intercept and numerator

Using the X-Intercept of Rational Function knowledge point

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Determine constant using y-intercept

Using the Y-Intercept of Rational Function knowledge point

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Answer:

• A. \(f(x) = \frac{20}{x + 10}\)
• B. \(f(x) = \frac{20}{x - 10}\) (Correct answer)
• C. \(f(x) = \frac{10}{x + 10}\)
• D. \(f(x) = \frac{10}{x - 10}\)