Domain and Range of Rational Functions | General Mathematics | Prof D

0:00 hello class welcome to my channel in this video i will show you how to find the domain and range of the rational function so let's have example number one find the domain and range of the rational function f of x equals 1 over x minus 2. so a rational function is defined only when its denominator is not equal to 0. so this means that the domain of a rational function is the set of all real numbers except those that make the denominator equal to zero so let's find the domain of the function one over x minus two so i'm gonna go in

0:46 and it guys is equal to the new denominator to zero then solve for the value of x so transpose nothing new negative two we have x equals two so you value the x number gig is zero in denominator nothing function is two so therefore and domain not in detail is the set of all real numbers such that x is not equal to two so then simply guys now so another example example we are asked to find the domain of x plus 5 over 3x squared minus 6x

1:33 okay so same procedure guys equate log nothing in denominator 2 0 then solve the value of x so in this case we solve nothing value x by factoring so factoring out in c 3x we have x minus two equals zero okay then equate both factors to zero we have three x equals zero and x minus 2 equals 0. so in this case we have two values of x naught into by 3 we have x is equals to

2:19 zero then your parallel we have x equals to 2. so therefore and domain uh thing g of x is the set of all real numbers such that x is not equal to zero and two all right so guys so next train number nothing range of the rational function f of x equals 1 over x minus 2 so u range guys on the other hand can easily be identified through its graph okay but it can also be determined

3:06 by replacing f of x by y and solving for x in terms of y so try nothing example this is our function so replace that is f of x to y equals one over x minus two okay then after that omega i and solve not in c c c x in terms of y so paramaga ion by cross multiplication alumni nothing but one denominator that is understood no one okay so y times x minus two equals one times one okay

3:54 then after an end solving for x divide both sides of the equation by y so maga cancel nothing then we have x minus 2 equals 1 over y then solving for x transpose 9 is negative two we have x equals one over y plus positive two okay so identifying the nut in c range as you notice the denominator so meaning but nothing is in debugging equals to zero

4:42 function so therefore arrange not n y is the set of all real numbers such that y is not equal to zero so again linka simply come over and arrange guys okay so next let's have example number two determine the domain and range of the rational function whose graph is shown below sodium guys is you observe nothing on graph so normally param identify not in c domain angular [Music]

5:37 plane so we have negative one negative two negative three there's a point now negative three horizontal line union point constant okay so therefore a domain non-acting graph is this x is the set of all real numbers such such that x is not equals to negative three okay so syringe and technically y values okay so considering the x titanium this time okay why don't

6:26 um vertical line okay then as you notice y is the set of all real numbers such that y is not equals to 1. so this is the end of our video if you have questions or clarifications kindly put them in the comment section below so thank you guys for watching this prof d i'll catch you on the flip side bye

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