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We have `y=f(x)=((x-1)(6x-1))/(2x-1)`. <br> The domain of the function is `R-{1//2}`. <br> 1. y-intercept `f(0)=1` So the graph cuts the y-axis at (0,-1). <br> 2. x-intercept (zeros) <br> Put `y=0` or ` (x-1)(6x-1)=0` <br> So the graph meets the x-axis at (1/6,0) and (1,0). <br> 3. Asymptotes <br> Vertical asymptotes <br> Clearly, the graph has vertical asymptote x = 1/2, where the denominator becomes zero. <br> Horizontal asymptotes Clearly, the graph has no horizontal asymptote. <br> Oblique asymptotes <br> `y=(6x^(2)-7x+1)/(2x-1)` <br> `=3x-2-(1)/(2x-1)` <br> Thus, important points and lines are as follows. <br> <img src="https://d10lpgp6xz60nq.cloudfront.net/physics_images/CEN_GRA_C06_S01_029_S01.png" width="80%"> <br> 4. Momotonicity/Extremum <br> `f'(x)=((12x-7)(2x-1)-2(6x^(2)-7x+1))/((2x-1)^(2))` <br> `=((12x^(2)-12x+5))/((2x-1)^(2)) gt0, AA x in R-{1//2}` <br> Hence the function is increasing throughout. <br> `underset(xrarr-oo)lim((x-1)(6x-1))/(2x-1)=-oo` and `underset(xrarr(1^(-))/(2))lim ((x-1)(6x-1))/(2x-1)=oo` <br> Thus, f(x) increases from `-oo` to `oo` when x increases from `-oo` to 1/2 crossing the x-axis at (1/6,0) and approaching asymptote `y=3x-2` to the left of it. <br> `underset(xrarr(1^(+))/(2))lim ((x-1)(6x-1))/(2x-1)=-oo` and `underset(xrarroo)lim ((x-1)(6x-1))/(2x-1)=oo` <br> Thus, f(x) increases from `-oo` to `oo` when x increases from 1/2 to `oo` crossing the x-axis at (1,0) . f(x) approaches asymptote `y=3x-2` to the right of the asymptote. <br> Hence the graph of `y=f(x)` can be drawn as shown in the following figure. <br> <img src="https://d10lpgp6xz60nq.cloudfront.net/physics_images/CEN_GRA_C06_S01_029_S02.png" width="80%">