Beginner calculus question requiring domain of a rational function in square root. Reddit r/calculus

beginner calculation requiring domains here we go here we have to find the domain for the function square root of 18x over x^2 minus 400 notice where we have a rational function instead of the square root and these are the conditions that we have to set up I would suggest you to work with the rational part first so whenever we have a top over bottom remember the bottom cannot be equal to zero so pay attention to the bottom and make sure that when Decay this right here cannot be zero so let's just put the bottom here whatever the expression is make it not equal to zero this way after you solve this or technically inequality because this is a not equal sign you end up with the numbers that we cannot plug in for this function next if you take a look at square root of the inside because we are just dealing with real numbers so when we have this case we have to make sure the inside of the square root cannot be negative numbers so that means we have to make sure the inside just write that down right the expression inside and then set it greater than or equal to zero and then solve for this inequality and because we have two situations we have to do both and then find out the interval that satisfy both conditions here we go let's go ahead and take care of the first part which is just the denominator of the rational part so imagine we don't have the square root just go ahead and do that so we will have to set X2 minus 400 not equal to zero and then just solve it I'm just going to move this to the other side right x² not equal to 400 and then we can just simply take the square root on both sides that way this and that cancel and and don't forget the plus or minus that implies X cannot be plus or minus 20 so when we have a Nal sign solving this it's the same as solving a regular equation and we found two numbers that we can use for this function now the harder part for this question is that we have to make sure the inside of the square root being positive so for the second part we have to write down the inside expression 18x over x^2 - 400 we have to make sure that is greater than or equal to zero so just kind of forget all that focus on this right here and this right here is also not easy to solve because we have a rational inequality or nonlinear inequality do not just multiply this on both sides it's not like that so how do we do it this is the safe way to do it go ahead and find out the numbers that we care especially the right hand side is zero already you can proceed to do that on the top we have 18 x make this equal to zero that implies X is equal to zero so if x is zero you'll get a number that we care right 18x equal z x can be zero so we first care about zero and the next you look at the bottom you also make it equal to zero and this implies it's the same thing right there X is equal to plus or minus 20 technically not equal because again they on the bottom so right here we have to care about zero and also plus or minus 20 and the best way to do this this is now take that and then draw a number line right here and then just test out interval we have to put in these three numbers right here so let say here is zero and then here is -20 and here is positive 20 remember we cannot have X is positive or negative 20 so right here I'm just going to put an open circle and open circle and because this is a greater than equal sign that means X could be equal to zero so zero is included so you can C put close Circle right here right that now test it out how pick a number less than ne2 so let's just go ahead and say - 21 don't be too fancy right let's say - 21 and then just plug into this right here so we have -21 and then squar minus 400 and then the top is 18 * -21 and you don't really have to work out like a specific numbers you can just test out if it's positive or not right here the top is positive oh sorry positive * negative is negative and then the bottom well this right here is going to be 440 something right 41 minus that so it's positive so work that out you get Negative over positive which Al together we get negative and that's not what we want because we're looking for the non- negative part so now go ahead and pick a number from between -2 and zero let's say one why not negative 1 because it's less than zero now Put negative 1 into all the xes we are looking at 18 * -1 over -1 Square - 400 now the top is negative the bottom right here is 1 - 400 is for sure negative all in all we get positive continue pick a number between Z and 20 that say 1 and then put it here so we're looking at 18 * 1 over 1^ SAR - 400 the top is positive one minus that is negative all all this right here is negative and we don't want that section lastly pick a number bigger than 20 that say 22 no don't be too fany just 21 plug in so we have 18 * positive 21 over 21^ 2ar - 400 the top is positive the bottom right here work that out you will get positive as well so all all you get positive so now this conditions are right here and right here already so we are good we can just kind of picked up what we need from here so ladies and gentlemen I will tell you the domain for the function is equal to we go from -2 to Z in interval notation we are not including the -20 so put on parenthesis but we are including the zero so so go ahead and put on the square bracket and then we have another part right here so you can put on Union and some of the computer softwares they will also accept comma doesn't really matter that much in my opinion so depending on which one that they take by Union usually the other one we have to go from 20 all the way to the right so all the way to Infinity for the 20 we are not including it so parenthesis and we can never include in Infinity so parenthesis and that is the domain for that function so hopefully this right here helps that's it

Channel: bprp calculus basics

Views: 14,072

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Id: qZSo1qwCZwM

Length: 8min 10sec (490 seconds)

Published: Sat Mar 16 2024