In the xy-plane, the graph of a rational function f has a vertical asymptote at x=-5. Which of the following could be an expression for f(x) ? (A) ((x-5)(x+5))/(2(x-5)) (B) ((x-4)(x+5))/((x-1)(x+5)) (C) ((x+1)(x+5))/((x-5)(x+2)) (D) ((x-5)(x-3))/((x-3)(x+5))

Question

In the  xy-plane, the graph of a rational function  f has a vertical asymptote at  x=-5. Which of the following could be an expression for  f(x) ? (A)  ((x-5)(x+5))/(2(x-5)) (B)  ((x-4)(x+5))/((x-1)(x+5)) (C)  ((x+1)(x+5))/((x-5)(x+2)) (D)  ((x-5)(x-3))/((x-3)(x+5))

Bytelearn · Accepted Answer

Identify Vertical Asymptote: Identify the characteristic of a vertical asymptote in a rational function. A vertical asymptote occurs where the denominator of a rational function is zero, but the numerator is not zero at the same x-value.Examine Option (A): Examine option (A) to see if it has a vertical asymptote at x=-5. The denominator of option (A) is 2(x-5), which is zero when x=5, not x=-5. Therefore, it cannot have a vertical asymptote at x=-5.Examine Option (B): Examine option (B) to see if it has a vertical asymptote at x=-5. The denominator of option (B) is (x-1)(x+5), which is zero when x=-5. However, the numerator also has a factor of (x+5), which would cancel out the (x+5) in the denominator, eliminating the vertical asymptote at x=-5.Examine Option (C): Examine option (C) to see if it has a vertical asymptote at x=-5. The denominator of option (C) is (x-5)(x+2), which is zero when x=5, creating a vertical asymptote at x=5. Since we are looking for a vertical asymptote at x=-5, this option is not correct.Examine Option (D): Examine option (D) to see if it has a vertical asymptote at x=-5. The denominator of option (D) is (x-3)(x+5), which is zero when x=-5. The numerator is (x-5)(x-3), which does not cancel out the (x+5) in the denominator. Therefore, this function has a vertical asymptote at x=-5.

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