Write a rational function f that has vertical asymptote at x=4, a horizontal asymptote at y=5 and a zero at x=−7.
A
f(x)=5(x−7)(x−4)
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B
f(x)=5(x+7)(x−4)
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C
f(x)=5(x−7)(x+4)
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D
f(x)=(x+7)(x+4)
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Solution
The correct option is Bf(x)=5(x+7)(x−4) Since the rational function f has the vertical asymptote at x=4, then the denominator of f contains the term (x−4).
Thus function f(x) is of the form f=g(x)x−4.
Since the horizontal asymptote exists y=5, the numerator g(x) of f(x) has to be of the same degree as the denominator with a leading coefficient equal to 5.
Also g(x) must contain the term (x+7) since f has zero at x=−7.