[tex]f(x)=\frac{3x-6}{5-2x}[/tex]Domain:V.A:Roots:Y-Int:H.A:Holes:O.A:Also draw on the graph.

Accepted Solution

i) The given function is [tex]f(x)=\frac{3x-6}{5-2x}[/tex]The domain is all real values except the ones that will make the denominator zero.[tex]5-2x=0[/tex][tex]-2x=-5[/tex][tex]x=2.5[/tex]The domain is all real values except, x=2.5.ii) To find the vertical asymptote, we equate the denominator to zero and solve for x.[tex]5-2x=0[/tex][tex]-2x=-5[/tex][tex]x=2.5[/tex]iii) If we equate the numerator to zero, we get;[tex]3x-6=0[/tex][tex]3x=6[/tex]This implies that;[tex]x=2[/tex]iv) To find the y-intercept, we put x=0 into the given function to get;[tex]f(0)=\frac{3(0)-6}{5-2(0)}[/tex].[tex]f(0)=\frac{-6}{5}[/tex].[tex]f(0)=-\frac{6}{5}[/tex].v)The degrees of both numerator and the denominator are the same.The ratio of the coefficient of the degree of the numerator to that of the denominator will give us the asymptote. The horizontal asymptote  is [tex]y=-\frac{3}{2}[/tex].vi) The function has no common factors that are at least linear.The function has no holes in it.vii) This rational function has no oblique asymptotes because it is a proper rational function.