Question

question
consider the graph of the function $f(x)=\frac{x^{2}+7x}{x^{3}-5x^{2}-14x}$.
what are the vertical asymptotes? just list the $x$-values, separated by commas.
provide your answer below:

Explanation:

Step1: Factor the numerator and denominator

The numerator $x^{2}+7x=x(x + 7)$. The denominator $x^{3}-5x^{2}-14x=x(x^{2}-5x - 14)=x(x - 7)(x+2)$. So, $f(x)=\frac{x(x + 7)}{x(x - 7)(x + 2)}$.

Step2: Simplify the function

Cancel out the common factor $x$ (for $x
eq0$), we get $f(x)=\frac{x + 7}{(x - 7)(x + 2)}$.

Step3: Find the vertical - asymptotes

Vertical asymptotes occur where the denominator is zero and the numerator is non - zero. Set the denominator $(x - 7)(x + 2)=0$. Solving $x-7 = 0$ gives $x = 7$, and solving $x+2=0$ gives $x=-2$.

Answer:

$-2,7$