Question

consider the graph of the function $f(x)=\frac{x^{3}-11x^{2}+30x}{x^{2}-5x}$. which is a removable discontinuity for the graph? select all that apply. select all that apply: $x = - 6$ $x = - 5$ $x = 0$ $x = 5$

Explanation:

Step1: Simplify the function

First, factor the numerator and denominator.
The numerator $x^{3}-11x^{2}+30x=x(x - 5)(x - 6)$.
The denominator $x^{2}-5x=x(x - 5)$.
So, $f(x)=\frac{x(x - 5)(x - 6)}{x(x - 5)}=\frac{x - 6}{1},x
eq0,5$.

Step2: Identify removable discontinuities

A removable discontinuity occurs when a factor in the denominator can be canceled out in the numerator - denominator expression.
The function $f(x)$ is undefined at $x = 0$ and $x = 5$ because these values make the original denominator zero. But since we can cancel out the factors $x$ and $(x - 5)$ in the simplified form, $x = 0$ and $x = 5$ are removable discontinuities.

Answer:

C. $x = 0$, D. $x = 5$