find equations for the vertical asymptotes, if any, for the following rational function.\n\n f(x)=\frac{12…

find equations for the vertical asymptotes, if any, for the following rational function.\n\n f(x)=\frac{12 x^{2}-10 x + 2}{-2 x + 1} \n\nanswer\nhow to enter your answer (opens in new window)\nseparate multiple equations with a comma.\nselecting a button will replace the entered answer value. the value of the button is used instead of the value in the associated box.

find equations for the vertical asymptotes, if any, for the following rational function.\n\n f(x)=\frac{12 x^{2}-10 x + 2}{-2 x + 1} \n\nanswer\nhow to enter your answer (opens in new window)\nseparate multiple equations with a comma.\nselecting a button will replace the entered answer value. the value of the button is used instead of the value in the associated box.

Answer

Explanation:

Step1: Find the denominator

The denominator of the rational function (f(x)=\frac{12x^{2}-10x + 2}{-2x + 1}) is (D(x)=-2x + 1).

Step2: Set the denominator equal to zero

Set (D(x)=0), so (-2x+1 = 0). Solve for (x): [ \begin{align*} -2x&=-1\ x&=\frac{1}{2} \end{align*} ]

Step3: Check if the numerator is non - zero at this (x) value

Substitute (x = \frac{1}{2}) into the numerator (N(x)=12x^{2}-10x + 2) [ \begin{align*} N(\frac{1}{2})&=12\times(\frac{1}{2})^{2}-10\times\frac{1}{2}+2\ &=12\times\frac{1}{4}-5 + 2\ &=3-5 + 2\ &=0 \end{align*} ] Factor the numerator (12x^{2}-10x + 2=2(6x^{2}-5x + 1)=2(2x - 1)(3x - 1)) Factor the denominator (-2x + 1=-(2x - 1)) So (f(x)=\frac{2(2x - 1)(3x - 1)}{-(2x - 1)}=-2(3x - 1)) for (x\neq\frac{1}{2})

Answer:

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