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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%">Transcript

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00:00 - 00:59 | restaurants in his brother graph of Y is equal to x minus 1 6 x minus 1 divided by 2 x minus 1 clearly domain is real number excluding point one by two because at x is equal to 1 by 2 this function is not defined so will get at x is equal to 1 by 2 will be the attempt to write function effects as effect will get 6 x square by multiplying 68 - 6 - 6 - 71 will get will get minus 1 whole square minus 1 whole square |

01:00 - 01:59 | differentiate numerator Bill Gate 12 x minus 7 into minus 1 minus differentiation of denominator will get to x square - 7 x + 1 after calculation what will get your leg 24 x square - 12 x 12 from here what will get 12 x minus 4 3 X - 2 - 12 - 14 - 26 - 26 + 40 12 concept will get 77 - 2 which is equal to 5 divided by minus one will of discriminant what we're getting this |

02:00 - 02:59 | is equal to 144 minus into 12 into 5144 warranty 12.25 which will get 144 -2 40 which is less than zero so clearly the coefficient of viscosity of this is going to be strictly for every real number is equal to 1 by increasing in the hole in trouble but it is not defined is equal to draw the graph by using this function is basically effect is given equal to x minus 16 - 1 x minus 16 x minus 1 divided by 2 x minus one |

03:00 - 03:59 | what is the value of getting - 1 - 11 / -1 which will give you -10 function value clearly if x is equal to 1 by 2 equal to 1 by 2 is your asymptote accessible 2122 is your Assam do so the graph will be the line asymptote will be something like that this line x is equal to 1 by 2 is getting zero effect is getting zero at x is equal to 111 is the year and other one by 6 |

04:00 - 04:59 | getting zero is equal to 1 by 6 1 by 6 is somewhat year this is why this is coin 1.1 by 6.1 by 6160 FX approaches to 1 by 2 dysfunction the grass is moving something like something like this and minus infinity limit of approaches to minus infinity of what we are getting dividing this will give you a project to minus infinity this will give you |

05:00 - 05:59 | X - 11 - 1 and 6 minus 1 by a divided into this is getting sick numerator is getting sick denominator is getting at 125 municipality this is getting getting 1266 this is getting getting close to do with negative people of the approaches will get this value will get negative |

06:00 - 06:59 | and this is getting negative this is also getting this is getting positive 1 by 3 1 by 2 is a substitute will get free -12 positive and this is getting negative 1 by 2 minus 1 this is negative and this is getting one by two this is getting negative negative negative negative positive approach class infinity and approaches to this is going to be positive this is positive this is negative moving to negative infinity and it will pass through some point and this will be |

07:00 - 07:59 | infinity by logic just denominator will approach 20 here with + X + Infinity so this is traffic this thank you |