i) The given function is
![f(x)=\frac{2x-1}{x^2-x-6}](/tpl/images/0490/6170/db23d.png)
We factor to obtain
![f(x)=\frac{2x-1}{(x-3)(x+2)}](/tpl/images/0490/6170/09934.png)
The domain is
![(x-3)(x+2)\ne0](/tpl/images/0490/6170/b5e3f.png)
![(x-3)\ne0,(x+2)\ne0](/tpl/images/0490/6170/1c137.png)
![x\ne3,x\ne-2](/tpl/images/0490/6170/663f7.png)
ii) The vertical asymptotes are
![(x-3)(x+2)=0](/tpl/images/0490/6170/e6713.png)
![(x-3)=0,(x+2)=0](/tpl/images/0490/6170/eb0c0.png)
![x=3,x=-2](/tpl/images/0490/6170/1e02f.png)
iii) To find the root, we equate the numerator to zero.
![2x-1=0](/tpl/images/0490/6170/ade47.png)
![x=\frac{1}{2}](/tpl/images/0490/6170/41833.png)
iv) To find the y-intercept, put x=0 into the function.
![f(0)=\frac{2(0)-1}{(0)^2-(0)-6}](/tpl/images/0490/6170/3eb7e.png)
![f(0)=\frac{-1}{-6}](/tpl/images/0490/6170/b6f26.png)
![f(0)=\frac{1}{6}](/tpl/images/0490/6170/45280.png)
vi) To find the horizontal asymptote, we take limit to infinity.
This implies that;
![lim_{x\to \infty}\frac{2x-1}{x^2-x-6}=0](/tpl/images/0490/6170/544a4.png)
The horizontal asymptote is y=0.
vii) The numerator and the denominator do not have common factors that are at least linear.
Therefore the function has no holes in it.
![domain: v.a: roots: y-int: h.a: holes: o.a: also draw on the graph.](/tpl/images/0490/6170/a91d8.jpg)