SOLUTION:
The function is;
[tex]r(x)=\frac{x^2-4}{x^2-3x+2}[/tex]
Factorizing the numerator and the denominator, we have;
[tex]r(x)=\frac{(x-2)(x+2)}{(x-1)(x-2)}[/tex]
There is a hole at x = 2.
Cancelling the common x - 2 terms, we have;
[tex]r(x)=\frac{x+2}{x-1}[/tex]
The hole exists at the point ( 2, 4 );
[tex]r(2)=4[/tex]
The y-intercept is the point (0, -2 ) ;
The x-intercept is at the point ( -2. 0) ; Since
[tex]\begin{gathered} x+2=0 \\ x=-2 \end{gathered}[/tex]
The horizontal asymptote is the line;
[tex]y=1[/tex]
The vertical asymptote is the line;
[tex]x=1[/tex]
The graph is plotted below;
Clearly, all attributes of the graph is evident here.
The domain and range of the function is;
[tex]Domain:(-\infty,1)\cup(1,2)\cup(2,\infty)[/tex]
The range is;
[tex]Range:(-\infty,1)\cup(1,4)\cup(4,\infty)[/tex]
