wing rational function, f(x) = (2x^2 - 4)/(x^3 - 8) equations for the vertical asymptotes, if any, for the…

wing rational function, f(x) = (2x^2 - 4)/(x^3 - 8) equations for the vertical asymptotes, if any, for the function. answer (opens in new window) equations with a comma, will replace the entered answer value. the value of the button is used instead of the value in the associated box, none

wing rational function, f(x) = (2x^2 - 4)/(x^3 - 8) equations for the vertical asymptotes, if any, for the function. answer (opens in new window) equations with a comma, will replace the entered answer value. the value of the button is used instead of the value in the associated box, none

Answer

Explanation:

Step1: Recall vertical - asymptote condition

Vertical asymptotes occur where the denominator of a rational function is zero and the numerator is non - zero. Set the denominator equal to zero: $x^{3}-8 = 0$.

Step2: Factor the denominator

We know that $a^{3}-b^{3}=(a - b)(a^{2}+ab + b^{2})$. Here, $a=x$ and $b = 2$, so $x^{3}-8=(x - 2)(x^{2}+2x + 4)=0$.

Step3: Solve for x

The quadratic factor $x^{2}+2x + 4$ has discriminant $\Delta=b^{2}-4ac=2^{2}-4\times1\times4=4 - 16=-12<0$, so it has no real roots. Solving $x - 2=0$ gives $x = 2$. When $x = 2$, the numerator $2x^{2}-4=2\times2^{2}-4=8 - 4 = 4\neq0$.

Answer:

$x = 2$