Question
Which function has a graph with a vertical asymptote at x = 3? Select all that apply .
![straight F left parenthesis straight x right parenthesis equals fraction numerator x minus 2 over denominator x squared plus 2 x minus 15 end fraction](data:image/png;base64,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)
![straight F left parenthesis straight x right parenthesis equals fraction numerator straight x minus 3 over denominator 12 plus 7 straight x plus straight x squared end fraction](data:image/png;base64,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)
![straight F left parenthesis straight x right parenthesis equals fraction numerator x squared minus 9 over denominator x plus 9 end fraction](data:image/png;base64,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)
- Non of the above
The correct answer is: ![straight F left parenthesis straight x right parenthesis equals fraction numerator x minus 2 over denominator x squared plus 2 x minus 15 end fraction](data:image/png;base64,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)
a) The vertical asymptote of a rational function is x -value where the denominator of the function is zero. Equate the denominator to zero and find the value of x .
x2 + 2x - 15= 0
x2 + 5x -3x - 15= 0
x(x + 5) – 3(x + 5)
(x + 5) (x - 3)
x = -5 and x = 3
The vertical asymptote of the rational function is x =-5 and x =3
b) The vertical asymptote of a rational function is x -value where the denominator of the function is zero. Equate the denominator to zero and find the value of x .
x2 + 7x + 12= 0
x2 + 4x + 3x + 12= 0
x(x + 4) + 3(x + 4)
(x + 4)(x + 3)
x = -4 and x = -3
The vertical asymptote of the rational function is x =-4 and x = -3.
c) The vertical asymptote of a rational function is x -value where the denominator of the function is zero. Equate the denominator to zero and find the value of x .
x + 9 = 0
x = -9
Function
a.
has vertical asymptote at x = 3.
x2 + 5x -3x - 15= 0
x(x + 5) – 3(x + 5)
(x + 5) (x - 3)
x = -5 and x = 3
The vertical asymptote of the rational function is x =-5 and x =3
b) The vertical asymptote of a rational function is x -value where the denominator of the function is zero. Equate the denominator to zero and find the value of x .
x2 + 4x + 3x + 12= 0
x(x + 4) + 3(x + 4)
(x + 4)(x + 3)
x = -4 and x = -3
The vertical asymptote of the rational function is x =-4 and x = -3.
c) The vertical asymptote of a rational function is x -value where the denominator of the function is zero. Equate the denominator to zero and find the value of x .
x = -9
Function
a.