find the values of x where the given function has vertical asymptotes. list them in increasing order. f(x) =…

Explanation:

Step1: Recall vertical - asymptote condition

Vertical asymptotes occur where the denominator of a rational function is zero and the numerator is non - zero. For the function $f(x)=\frac{1}{x^{2}-2x}$, we set the denominator equal to zero. $x^{2}-2x = 0$

Step2: Factor the denominator

Factor out an $x$ from the left - hand side of the equation: $x(x - 2)=0$.

Step3: Solve for x

Using the zero - product property, if $ab = 0$, then either $a = 0$ or $b = 0$. So, $x=0$ or $x - 2=0$. Solving $x - 2=0$ gives $x = 2$. The numerator is a non - zero constant ($1$), so these are the vertical asymptotes.

Answer:

$x = 0$, $x = 2$