find the equation of all vertical asymptotes of the following function.\nf(x)=\frac{2x - 8}{sqrt{48 - 6x}}

Explanation:

Step1: Recall vertical - asymptote condition

Vertical asymptotes occur where the function is undefined and the denominator approaches 0 while the numerator does not. Set the denominator equal to 0. $\sqrt{48 - 6x}=0$

Step2: Square both sides

Squaring both sides to get rid of the square - root, we have $48 - 6x = 0$.

Step3: Solve for x

First, add $6x$ to both sides: $48=6x$. Then divide both sides by 6. So, $x = 8$. We also need to consider the domain of the original function. The expression under the square - root must be non - negative, i.e., $48-6x\geq0$ or $x\leq8$. When $x = 8$, the numerator $2x - 8=2\times8 - 8=8\neq0$.

Answer:

$x = 8$