[tex]f(x) = \frac{2x-1}{x^{2}-x-6 }[/tex]Domain:V.A:Roots:Y-Int:H.A:Holes:O.A:Also draw on the graph.

i) The given function is [tex]f(x)=\frac{2x-1}{x^2-x-6}[/tex]We factor to obtain[tex]f(x)=\frac{2x-1}{(x-3)(x+2)}[/tex]The domain is [tex](x-3)(x+2)\ne0[/tex] [tex](x-3)\ne0,(x+2)\ne0[/tex] [tex]x\ne3,x\ne-2[/tex] ii) The vertical asymptotes are [tex](x-3)(x+2)=0[/tex] [tex](x-3)=0,(x+2)=0[/tex] [tex]x=3,x=-2[/tex] iii) To find the root, we equate the numerator to zero.[tex]2x-1=0[/tex][tex]x=\frac{1}{2}[/tex]iv) To find the y-intercept, put x=0 into the function.[tex]f(0)=\frac{2(0)-1}{(0)^2-(0)-6}[/tex][tex]f(0)=\frac{-1}{-6}[/tex][tex]f(0)=\frac{1}{6}[/tex]vi) To find the horizontal asymptote, we take limit to infinity.This implies that;[tex]lim_{x\to \infty}\frac{2x-1}{x^2-x-6}=0[/tex]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.