The vertical asymptotes of a function are the zeroes of the denominator of a rational function
The function is given as:
[tex]\mathbf{f(x) = \frac{(x-9)}{(x^3-81x)}}[/tex]
Set the denominator to 0
[tex]\mathbf{x^3-81x = 0}[/tex]
Factor out x
[tex]\mathbf{x(x^2-81) = 0}[/tex]
Express 81 as 9^2
[tex]\mathbf{x(x^2-9^2) = 0}[/tex]
Express as difference of two squares
[tex]\mathbf{x(x-9)(x + 9) = 0}[/tex]
Split
[tex]\mathbf{x = 0\ or\ x-9 = 0\ or\ x + 9 = 0}[/tex]
Solve for x
[tex]\mathbf{x = 0\ or\ x=9\ or\ x =-9}[/tex]
So, the vertical asymptotes of f(x) are 0, 9 and -9
See attachment for the graph of f(x)
Read more about vertical asymptotes at:
https://brainly.com/question/4084552
