Answer:
The correct option is a.
Step-by-step explanation:
The given function is
[tex]f\left(x\right)=\frac{\left x^2-81\right}{x^2-11x+18}[/tex]
It can be written as
[tex]f\left(x\right)=\frac{\left(x-9)(x+9)}{(x-9)(x-2)}[/tex]
[tex]f\left(x\right)=\frac{\left x+9}{x-2}[/tex]
Put x=0 to find y-intercept.
[tex]f(0)=\frac{0+9}{0-2}=-4.5[/tex]
The y-intercept is (0,-4.5).
Put f(x)=0 to find x-intercept.
[tex]0=\frac{x+9}{x-2}\Rightarrow x=-9[/tex]
The x-intercept is (-9,0).
Equate denominator equal to 0, to find the vertical asymptote.
[tex]x-2=0\Rightarrow x=2[/tex]
The vertical asymptote is x=2.
Take limit x tends to infinity, to find horizontal asymptote.
[tex]lim_{x\rightarrow \infty}f(x)=lim_{x\rightarrow \infty}\frac{x+9}{x-2}[/tex]
[tex]lim_{x\rightarrow \infty}f(x)=lim_{x\rightarrow \infty}\frac{x(1+\frac{9}{x})}{x(1-\frac{2}{x})}[/tex]
Apply limits,
[tex]lim_{x\rightarrow \infty}f(x)=1[/tex]
The horizontal asymptote is y=1.
Since the intercepts and asymtotes lie in the window, i.e., Xmin: –10, Xmax: 10
, Ymin: –10, Ymax: 10, thus the correct answer would be option a.
