Determining Domain of Functions
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Determining Domain of Functions

HELLO again. PreCalc lesson number four of
the day or something like that. So let’s talk about determining the domain of a function
without the use of a graphing calculator. Now domain is the x’s that a graph can take
on. Well, equations makes those graphs. If you want to find domain without the, let’s
call it a crutch of a graphing calculator, you need to look at the equation itself. There
is really only two problems or issues when you are looking for the domain of an algebraic
equation or function. Now unless of course you are talking about trig functions which
are a whole different story. That is chapters from now, but sine, tangent, are a little
bit different. But when you are looking at a pure algebra equation at this level of math,
precalculus we are looking for the fact that you cannot divide by zero and you cannot take
the even root of a negative number. So let’s run through quite a few examples here in the
next fifteen minutes, so I have got about 15 minutes. Let’s see how many I can get through
before our time runs out. Is there an issue with the domain of f of x equals three x squared
plus four x minus seven? Well, the answer to this question is no. There is not a problem
with domain. The domain is going to be your x’s from negative infinity to positive infinity
and beyond. The reason why is because…ok. Let’s talk about… We are not going to break
out a graphing calculator, but when you have one power of two is that not a parabola. And,
when x is squared your parabola will open up or down, and therefore it will move left
and right forever. So, the domain is all real numbers. Let’s take a look at another function.
You know what I like about chalk boards so much even though they make me a mess? A box
with 12 pieces of chalk costs a dollar which is way less than dry board markers. You know
what else I like about chalk boards for? Because you can make the kids cringe at the slightest
drop of a hat. I love that chalk board sound:) Anyway. Let’s take a look at the next problem.
We have f of x is equal to two x squared plus six x plus four over x plus one. Now, what
is the problem here? Right away we can look at this and see that we have x in the denominator.
X in the denominator? If x is in the denominator, that is a problem. You cannot divide by zero.
So let’s see if that denominator can be made to equal zero. Well, clearly if I just subtract
both sides by one, x cannot equal negative one. So my domain is going to be all my x’s
from negative infinity to negative one, I can skip over that, and I can pick back up
on the other side of negative one and go to positive infinity. OR BEYOND! So why am I
leaving you this extra space here? Well, actually I don’t think that I left enough. Let’s see
if I can fit this in. The top, I want to simplify this function here and show you sometimes
the domain issues, or the domain restrictions, kind of disappear but you still need to pay
attention to them. A lot of times when the denominator equals zero, or can be equal to
zero, you have some form of asymptote, whether it is a vertical or a slanted asymptote. With
this though, we are not going to be left with this denominator. If I factor the numerator,
which is going to be… Well, you might want to see that reviewed. I just did this in another
video, I think I used this exact same function but let’s see. I told you that you could take
the first and last coefficient/term, multiply them together. This is my scratch work on
the side. Look for factors of that product that add to the middle term. Well, the middle
term is 6x. So if I use two and four. This is the same example if you just watched the
other lesson. Two x squared, I am going to use these and put x’s next to them. I am using
the two and four because they add up to six. So, plus 2x from there plus 4x from there
plus four all over x plus one. Now with these four terms, the way that I have split up the
six in the middle, I can do factoring by grouping. These two terms both share a 2x. 2 x squared
divided by 2x is x. 2x divided by 2x is one. Then these both have a common factor of four,
so we factor that out. Now look. We have two big terms which also both have a factor of
x plus one. I can take that out and write the x plus once instead of twice, leaving
me with 2x plus four all over x plus one. Guess what? Our denominator cancelled out.
Now if you think 2x plus four is the answer, you would right. But, the 2x+4 implies that
the domain is all real numbers because there is no division left. That undefined value
of negative one, actually if you looked at the graph would be a hole at negative one
and not a vertical asymptote. But that is beyond the scope today’s lesson, other than
we just want to make sure that x cannot equal negative one for this domain. So, let’s take
a look at the next problem, shall we??? The next problem I would like to go over with
you today is f of x is equal to the square root of x squared plus three. Well now you
go, woooh. you just told me a second ago I cannot take the even root of a negative number.
So what is inside has to remain positive. X squared plus three has to remain positive.
Well how good are you with your math? You take a number and you square it, it stays
positive. Then you add three and it is still positive. So there is no way that this will
come out to be less than zero so the domain is all real numbers. But if you don’t catch
that and you are actually trying to solve it you are going to bring three over to the
other side. And, if you still don’t catch it you will try and square root both sides
and haha! At some point you are going to realize that you cannot square root a negative number
unless you are working with imaginary numbers. Right now, we are not. So, there is no problem
with the domain. This will never be negative, so I can plug in anything that I want. If
I can plug in anything that I want, then the domain are our x’s from negative infinity
to positive infinity AND BEYOND!!! Alright, so moving on to the next example. Let’s take
a look at a square root that will have a limit on the domain. So, the square root of x square
minus four. Ok, nice four. That is what I get for looking forward while I am writing.
Once again x is in the square root symbol. So, what is inside the square root symbol,
the even root, has to remain positive. So, x squared minus four is greater than or equal
to zero. That means that x squared must be greater than or equal to four after you add
the four over to the other side. Then finally we got that power of two. We need to undo
that. I am going to square root both sides and don’t forget you have both a positive
and negative answer. So, x has to be greater than or equal to two or x has to be, I am
changing the direction of the sign for the negative answer. Don’t forget that with even
powers and you are working with inequalities. x has to be less than or equal to negative
two. So there is my domain. X has to be greater than or equal to two or less than or equal
to negative two. Negative two? uh. So what does that mean? Just means what it says. From
negative infinity up until negative two, and it can be equal to because I can square root
zero, or you can use values that start at two and go to infinity. Of course we all know
that is …AND BEYOND!!! So if you plug in say three, three squared is nine, and nine
minus four is positive. You can take the square root of it. If you plug in negative three.
Negative three squared is positive nine, not negative nine, and positive nine minus four
is also going to be positive so BOOOM! There you go, these values will work in this equation.
Domain. Alright. I don’t hear my warning bell going off, so that means that i have time
for more examples. Oh, let’s see what else. F of x is the square root of x plus five over
x plus two. Well, now this equation gives us both problems. We can’t divide by zero
and we can’t take an even root of a negative number. We gotta check both of them if you
have both issues in your equation. X cannot equal negative two. Now the numerator. Let’s
highlight that. X plus five must remain greater than or equal to zero. So, when you solve
that inequality x has to remain greater than or equal to negative five. Now x can be greater
than or equal to negative five, but negative two is also bigger than negative five and
you cannot use that value. So, our final domain is your x’s such that it is from negative
infinity up until negative five. OOOHH, no it is not. NO, NO, NO…. Negative five is
the smallest x I can use, sorry about that. So, negative five, I got infinity stuck in
my head, negative five up until we get to a value of negative two. And then I can skip
over that two because I can’t use exactly negative two. Pick back up on the other side
of negative two and go to infinity….and beyond…yes I know. Alright, so there is
my domain for this function. You have to check the radical if it has an even root and you
have to check that you do not divide by zero. There you go. These numbers I can plug into
this function and not make it undefined. Alrighty! I think I have time for one more example and
I won’t scratch my nails down the board. I will just finish with a big ol’ BAM!!! That
is f of x is equal to the square root of x minus seven plus the square root of negative
x plus two. Let’s see how this works. Again, even roots. Let’s put a four in here. I am
tired of writing square roots. Even roots. You can’t even root a negative number. So,
once again this has to remain greater than or equal to zero. So, x minus seven has to
be greater than or equal to zero and negative x plus two has to remain greater than or equal
to zero. What do we have? This says that x has to be, after you add both sides by seven,
greater than or equal to seven. Here we are going to need to subtract by two. Negative
x is greater than or equal to negative two. We are going to divide by negative one. I
am going to show that step because when you divide by a negative number, please remember
to change the direction of your inequality. We get x is less than or equal to two. Ok,
so I have to meet both of those restrictions if I am going to be find the domain, or excuse
me, work out this equation. Both of these radicals are in the same equation, so I have
to meet both requirements. Whoo.. Well, I can’t!


  • ProfRobBob

    I tried to explain how it could look like -1 could be included in the domain, but you a correct in that it is not in the domain of the original function. The graph would have a hole at x = -1.

  • ProfRobBob

    THANK YOU for the spectacular review!!! Thank you too for subscribing and your support. My channel growth comes from subscribers like you "liking" and spreading the word.

  • gary warren

    I passed algebra with an A , I am now passing Trig with an A, and its all because of you Mr Tarou ! I am now brushing up on College Algebra using your videos for my Entrance math exam into Missouri S&T. You are truly the best, but I am with McLovinKatrina , PLEASE no more scratchin the board ! lol

  • Anna Anita

    I did not mind the scratch on the board, (I used to do it in high school to annoy others lol)…but I did not like that youtube only gives 15 minutes:(

  • ProfRobBob

    Yeah, you have learned how much I like to explain how things work in my videos and 15 minutes was tough to work with…for me!!!

  • MrSomeone0001

    Thanks you so much sir. I'm an internationla student and this videos will be helping me alot again thanks you so much 🙂

  • ProfRobBob

    You're welcome…thanks for subscribing and choosing my channel to learn from! I hope you continue watching and sharing my channel:D

  • Namrata Chauhan

    Sir, you are awesome!! I wish I could fly away from India to your country. Haha… finding out the domain of a function is so easy for me now. 🙂

  • ProfRobBob

    And that's awesome to hear…I love seeing all of my videos that you have watched and liked.
    I too wish you could fly over from India to be one of my students…school starts next week and I can never have too many motivated students in my classroom:) I'll look forward to updates throughout the school year!

  • IOW

    LOL ive been watching you since high school and now my calculus teacher is using your videos HAHA that gave me a good laugh, the link she asked us to watch was this video omg thats so awesome ^^

  • MsTwte

    instead of doing multiple examples of the same things maybe you could go into what happens if you have for example

    ( 2x -3 if x less than0
    f(x) ( x+1 -4 if x greater than 3
    ( 2x-4 if x less than 3

  • Christian Castillo

    Definitely going to keep referencing your channel for my precalfulus class and other math classes for when I get to college! Keep up the good work

  • ProfRobBob

    HAHA! Because I am one of the few teachers that still can and many of my students have never heard nails on a chalk board. Heck, some of my students have never seen a chalk board in a classroom.

  • Vivian De Almeida

    Hey ProfRobBob you've helped me understand Algebra 2 way better than my Algebra 2 teacher, so thank you very much. I recommended your channel to a bunch of other people on my teacher's roster who also are having a hard time. You rock at tutoring, so thanks for doing so.

  • ProfRobBob

    FIRST- Sorry for the delay in my reply…I have all comments set for approval and never saw this till I was answering another comment on this page:(
    Thanks for choosing my channel to watch and learn from and I really appreciate your support…I hope you will also share my channel with others who might benefit and remind them that it is important to like and SUBSCRIBE to help these free educational channel groW!

  • ProfRobBob

    You are so very welcome! And I really appreciate all your additional recommendations! Educational channels like mine need this kind of support and everyone watching to like and SUBSCRIBE in order to help us groW!
    Building a channel and teaching full-time is a lot of hard work but the reward of helping so many students like yourself makes it all worthwhile! Thanks for choosing Tarrou's Chalk Talk!

  • Derpboy Zero

    Only my second video in, and I'm learning the same material my Algebra 2 Honors teacher taught, but much, MUCH easier to understand with your instruction! She's so boring, and often screws up in her math making it even harder to learn.
    My teacher should have put your videos in the syllabus to refer to per lesson, so thanks Rob!! 😀

  • LeonDon

    Great video. Thanks once again… Your channel was very helpful last semester for my Trigonometry class, and now is being helpful again for my precalculus. 

  • Quark

    I am a relatively poor college student, starting on my journey to be a particle physicist. I have been watching your videos for well over a year now, and I wouldn't have gotten half as far as I am now without your help. You have been my teacher, and my friend. If I ever have the opportunity, I'd like to shake your hand and buy you a drink. I'll be donating more in the future, please keep the videos coming.

  • Rave Meyer

    Domain (f+g)(x); f(x)= 2 sqrt x-1  +7 ; g(x) 1/x
    Help?  I understand that a sq rt cannot be negative so in this case it might be x >= 1, but what about the rest of the equation, how do you treat that?  2 sqrt x-1 +7   do you just forget they're there?

  • Asa P

    I really appreciate your videos and time to help us struggling in math. Your energy and passion definitely motivates my willingness to learn. Thank you


    hey thnxs so much for your clases i´m learning a lot been studying administration but i got math lesons but my teachers are not even half you are. hopefully y can speak english. hahhaha. bucause i´m from Perú so i speak spanish but i learn english so make me easy follow your clases thnxs. for all what you do for people like me and others that we are nor good in maths. thnxs again. and keep going…..

  • Shivangi Singh

    @ProfRobBob I have a doubt. Why do we believe that anything in the underroot cannot be negative, when we know for a fact that both the squares of 2 & (-2) give us a 4, which in turn implies that the square root of 4 is ±2 ? Maybe I'm wrong somewhere but I will really appreciate it if your clear my doubt 🙂

  • Alpha224 T V

    MR ROB may y make some more videos about giving the domain of peace wise function that are more complicated please???? at the end of this month I will have a test save my life again please.  

  • 지형구

    Thanks for video! I am South Korean, but have some problems with Math in college 🙁 
    Your video helped me a lot 

  • Gerardo Gutierrez

    Please help!!

    I need help finding the domain of problems like

    So logs and other problems ive never seen before for my summer ap calc hw

  • Stephen

    Just to make sure, when I determine the domain of a function I use "Set Notation." 
    Which way is better to use… Set Notation or Interval Notation?

  • Dania

    In 8:09 Why is one X greater than 2 and the other X less than 2? Please answer I have a final coming up I would really appreciate the help

  • Fearsome hero

    ProfRobBob Great! My rare occurrence of facial ticks where triggered by you scratching on the chalkboard!
    No hard feelings! It was pretty funny. Plus I now have a good reason to take a break 🙂

  • Chris Wong

    RIP for the people who used headphone or earphone. Please turn the volume down when he is going to say "OR BEYOND" ! Anyway, won't tell ya when.

  • Stanley Rustling

    at like 9:03 why do you use this bracket if the number isn't inclusive I was always taught that parentheses were used when the numbers were non inclusive and brackets were used when they were

  • Haute Swan

    So helpful. If I watch it twice I should have it down pat. Thank you. When I am in between school sessions I can help you with closed captions but I have no clue how to do it though I am a computer wiz.

  • Ari Markou

    Hello I'm having trouble with pre-calc this year and most of it has to do with not understanding how the domain works. if it is undefined will the domain always be the number followed with infinity?

  • Harry S.

    Hey Mr. T, I have said this on previous videos, but you are just the best! Hard to express in words how helpful these videos are and how much I, along with 99% of the other people who watch these videos, appreciate everything you do!!!

    P.S. I got a 97 total on my trig final because of your great videos ?

  • Lughmani Tehreem

    I came back for some revision after two years, I know that cause I saw my comment from two years ago

  • Florin C.

    Thanks for this video and all your other videos. You helped me get an A in College Math. Now I'm having my kids watch your videos and get more ideas on solving math! Thanks.

  • Alye kaba

    its wired but i like it each time he corrected himself -mistakes makes him more human and I can relate to him. Great job..!!

  • Sabrina Nguyen

    Hello Professor! Thank you for this video! @10:20 what is the reason that negative infinity wasn’t included in the domain?

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