 So in the previous video as we talked about algebraic functions and how if we're given x we can find y and if we're given y How we can find x in the algebraic setting now an important question related to you know finding x and y coordinates is What is the total set of x coordinates? What is the total set of y coordinates? That is how does one find the domain in range of an algebraic function? Well, it turns out the answer about range is a little bit tricky It kind of pins on the family of functions and as we study different families of functions this semester We'll be able to answer questions about About range forthcoming rides of polynomials. We apply one thing radicals will say another thing exponentials bill say another thing So we'll approach questions about range on a future day. That's that's a little bit more sophisticated for Two too too much for us right now But we are in a position where we can talk about the domain of a function And so when a function is defined algebraically, we're gonna follow the following domain convention If a function is given by an algebraic expression and the domain is not stated explicitly So this would be like saying something like well f of x equals 2x plus 5, you know They tell you this is the function, but they don't specify what the domain is If ever I want to specify the domain I can tell you like oh, okay The domain is when x is greater than equal to 0 I can make that Specification or I might say something like the domain of f is Equal to 0 to infinity same thing right there. I could specifically tell you it But if it's not told if the domain is implicit that is we're just supposed to infer from the formula What the domain is we follow the following convention the convention is that the domain is the set of all real numbers For which the expression makes sense and defines a real number in its unsimplified form And so if a function of course is given by a graph or table then the domain is All those numbers represented on the graph or table. We've talked about that before but for an algebraic function We have to then infer what makes sense here for given the formula So assuming x is a real number. Is there a real number that we cannot multiply by 2? The answer is no we can multiply any number by 2 Can we add 5 to any number we can and so as such there's no restriction on the real number x right here so the domain would be all real numbers which an interval notation You'll write this as negative infinity to infinity right here or as a shortcut for that you can write this double r This is short for all real numbers and for many functions. That's what it's going to be We want the domains to be as big as possible And so we just want to infer what's the largest doing possible given this formula now at this time in the course There's only going to be two Concerns that are going to show up that might restrict the domain of an algebraically defined function though We some more we add later, but for the moment We have concerns of the following nature if you take the square root of negative one or in fact if you take the square root of any Any negative number this actually gives you an imaginary number. We'll talk about imaginary numbers more in the future When we have we will have a lesson about complex numbers. What have you but if you get an imaginary number I should mention that this is not a real number and so this would Violate the domain convention because the domain convention says that the number that comes out has to be a real number So if you take the square root of negative, that's not going to be a real number So it's not going to work. We'll we might relax that condition later, but for now Another problem you have to look out for is division by zero Now unlike the complex number system division by zero really cannot be resolved The consequences of allowing division by zero are too strenuous. It would destroy our algebra I'm not kidding about that. It would really would devastate the algebraic system We can't allow division by zero and I want to give you just a sort a short little argument Why that was suppose we did allow division by zero. Let's pick our two favorite numbers, right? I'm looking at the clock We are four minutes into the video and today is let's see my calendar is September 1st So I'm gonna prove to you. We're gonna prove That four equals one, right? And so here's here's the proof of that statement start off with something which we know is true zero equals zero great No one could deny that one Next well zero is the same thing as zero times four because anything times zero is four Sorry anything anything times zero zero right and zero is also zero times one Great, so still true nothing wrong with that, right? but take this equation again and If we were allowed to divide by zero then we could divide both sides by zero and we would get We would get Four equals one la la la la and then the arguments over with right So we've now proven that four equals one using this erroneous arithmetic of Division by zero if we allow division by zero we would have to we would have to allow that four equals one But why just stop there? We would have that any number is equal to one We have any numbers equal to zero and now we have a very depressing number system Wouldn't we you wake up at zero hundred hours in the morning you look at your bank account You have zero dollars you go on Facebook You have zero friends you have to drive to school and you're going zero miles per hour It would be a very impractical number system where every number is zero So if we want there to be non zero numbers We cannot allow division by zero and it's because of this dominance property of zero when you multiply by zero You always get back zero and because of that we can't negate multiplication of zero which is what division is all about So look out for division by zero and so using the domain convention Let's look at the following functions and determine what their domains would be So when you look at this first one, this is an example of a polynomial function You have some multiplication five times x exponents x square which just means x times x you have addition And so this doesn't have any of the problems. We we have to worry about there's no square roots involved There's no division and so by default your domain the domain of f right here is going to be all real numbers Negative infinity to infinity written in interval notation. That's all there is to it Now with example b right here the function g of x is defined as the square root of x plus 2 Because this is a square root function We do have to be cautious that make sure that x plus 2 is not negative And so because the radicand that's the expression inside of the radical there It's not an evolution of radicator or anything like that. The radicand has to be non negative So we have to solve the inequality x plus 2 is greater than equal to zero We'll talk some more about solving inequalities in the future, but this one's pretty mild It's very much like solving a linear equation You just have to remember to flip the sign if you ever multiply or divide by a negative So subtracting 2 from both sides we end up with x needs to be greater than equal to negative 2 Now this is greater than or equal to notice if you take x to be negative 2 G of negative 2 is actually equal to the square root of negative 2 plus 2 which is square of 0 There's no worries about taking the square root of 0 the square root of 0 is itself 0 Which is a real number negative 2 is inside the domain and in fact the domain of g right here is going to be an interval notation negative 2 to infinity and make sure that negative 2 is included in that you want to bracket Because evaluating the function negative 2 does give us a real number there So when you have square roots, you're gonna have to make sure that all radicands present are Non-negative on example c right here We have x over x squared minus x because this is a fraction in fact This is what we call a rational function. It's a ratio between two polynomials We have to make sure that the denominator is not equal to 0 That's the thing that gives us pause right here And so by doing this we basically want to solve the equation x squared minus x is not equal to 0 But of course solving the equation not equal to 0 is basically the same thing as solving equal to 0 So if you want to solve the equation equals 0, that's fine You'll you know remember that the answers you find will then be the things not in the domain So if you want to solve this one, uh factory makes a good technique It is a quadratic polynomial So you could like complete the square or use the quadratic formula Again, these are all techniques that we will review in the future as we study in depth more about quadratic equations But it's very likely you've seen some of these things before By a very simple factoring technique you can just factor out the greatest common divisor between x squared and x You can factor out the x that leaves behind an x minus 1 And that the only way that a product of two real numbers is going to equal 0 is if one of the factors was 0 This would give us that either x equals 0 or x minus 1 equals 0 Which solving the second equation tells us that x equals 1 These are the numbers 0 and 1 will make the denominator go to 0 And these are the only numbers that will make the denominator go to 0 And therefore these are the only numbers that are forbidden from the domain So the domain of h we might say something like the following We want all real numbers x such that x is not equal to 0 or 1 We can describe the domain using the set builder notation Because it's much easier to list the two exceptions than it is to list every number that's allowed But it but an interval notation this would look like negative infinity up to 0 parenthesis union 0 to 1 Again, parentheses here and then union 1 to infinity You could write an interval notation interval notation that way And we're just saying we want all real numbers except for 0 and 1 Do make sure you put parentheses by the 0 and the 1 because they are not included in the domain Because it makes the denominator go to 0 and also make sure you put parentheses by infinity and minus infinity because as those are real numbers They cannot be included inside of the domain only numbers are allowed there real numbers And so as one last example of this domain convention Let's find the domain of the function capital f of x equals the square root of 3x plus 12 Over x minus 5 Now one thing I do want to mention that you'll notice here This is capital f as opposed to the lowercase f when it comes to mathematics Mathematics is case sensitive if I write a capital f that does not mean lowercase f And if I were a lowercase f that does not mean capital f In this situation really wouldn't matter if I call it lowercase f But make sure that you do use capitals when you're supposed to and lowercase was when you're supposed to Because there are many situations which the lowercase and uppercase letter of the same Same alphabetic letter could show up in the formula together and those mean two different things So do pay attention to that Now with this function d capital f right here for example d there's two potential problems The first one comes from division by zero, right? So if you look at the denominator, you have this x minus 5 We have to figure out when x minus 5 equals zero will add 5 to both sides. You get x equals 5, right? And so this is the exception. We don't want x to equal 5. That's one problem with the domain Another potential problem from the domain comes from the radical in the numerator, right? We had the square root of 3x plus 12 So that concern means that if we're going to be a well-defined real number We need to take 3x plus 12 and set it gradient equal to zero and solve that equation minus 12 from both sides We'll write that out there. So we want us to track 12 from both sides The 12s cancel on the left hand side This gives us 3x is greater than equal to negative 12 divide both sides by three The threes would cancel on the left hand side. We then get that x is going to be greater than or equal to negative 4 And so we have sort of two things we we have to consider in order to get the domain x has to be gradient equal to negative 4 to make the numerator a real number And the denominator cannot be 5 Otherwise we divide it by zero and so when we put these things together, right? We have to take we have to take the intersection of these things, right the intersection We have to look at the overlap between x is greater than equal to negative 4 and x doesn't equal 5 And so this tells us that the domain of capital f is going to be negative 4 Up to 5 parentheses. So we have a bracket on the negative 4 a parenthesis on the 5 union 5 to infinity We're again, we have a parenthesis on the 5 that's because 5 is not allowed inside the domain But negative 4 is allowed in the domain because negative 4 would make the numerator go to 0 And it would make the denominator go to negative 9 0 over negative 9 is 0. That's that's a number there That's that's not a problem. And so this shows us how we can determine the domain of an algebraically defined function And so square roots of negatives and division by zero the things we have to look out for Like I said, there will be some problems that will appear in the future as we learn about some other function families But for a good while these are the only ones we have to worry about as we try to determine the domain of an algebraically defined function