i) The given function is
![f(x)=\frac{x-5}{2x^2-5x-3}](/tpl/images/0317/0904/55e35.png)
The factored form is
![f(x)=\frac{x-5}{(x-3)(2x+1)}](/tpl/images/0317/0904/6b468.png)
The domain are the values of  x for which the function is defined.
![(x-3)(2x+1)\ne 0](/tpl/images/0317/0904/eee07.png)
![(x-3)\ne0,(2x+1)\ne 0](/tpl/images/0317/0904/a6734.png)
![x\ne3,x\ne-\frac{1}{2}](/tpl/images/0317/0904/21638.png)
ii) To find the vertical asymptotes, equate the denominator to zero.
![(x-3)(2x+1)=0](/tpl/images/0317/0904/90fdc.png)
![(x-3)=\ne0,(2x+1)=0](/tpl/images/0317/0904/53bbe.png)
![x=3,x=-\frac{1}{2}](/tpl/images/0317/0904/de47e.png)
iii) To find the roots, equate the numerator to zero.
![x-5=0](/tpl/images/0317/0904/a997d.png)
The root is ![x=5](/tpl/images/0317/0904/4da05.png)
iv) To find the y-intercept, put
into the function.
![f(0)=\frac{0-5}{(0-3)(2(0)+1)}](/tpl/images/0317/0904/2d120.png)
![f(0)=\frac{-5}{(-3)(1)}](/tpl/images/0317/0904/a2e18.png)
![f(0)=\frac{5}{3}](/tpl/images/0317/0904/82495.png)
The y-intercept is ![\frac{5}{3}](/tpl/images/0317/0904/dcb5a.png)
v) The horizontal asymptote is given by;
![lim_{x\to \infty}\frac{x-5}{2x^2-5x-3}=0](/tpl/images/0317/0904/b3f09.png)
The horizontal asymptote is ![y=0](/tpl/images/0317/0904/08c08.png)
vi) The function is not reducible. There are no holes.
vii) The given function is a proper rational function.
Proper rational functions do not have oblique asymptotes.
![domain: v.a: roots: y-int: h.a: holes: o.a: also, draw on the graph.](/tpl/images/0317/0904/cc870.jpg)