 Hi, this is a video about rational functions finding the domain and using transformations to graph First and foremost, what is a rational function? by definition rational functions of the form They have your function r of x equals p of x over q of x where p and q are polynomial functions and Q of x is not equal to zero so pretty much what we do is we create a fraction from two polynomials and That's a rational function So remember you cannot divide by zero, which is why the bottom function or what's in the denominator cannot equal zero So I provide a few examples here such as r of x equals 2x to the fourth minus four over x plus five R of x equals x cubed over x squared plus one the only absolute necessary requirement to have a rational function is that the variable x must be in the denominator So rational functions are fraction functions So now with regards to the domain the restriction is that the denominator cannot equal zero So let's find the domain of each function in our first example. We'll give the answer and set builder notation So r of x equals 2x to the fourth minus four over x plus five So we know that x plus five is not allowed to equal zero because you cannot have zero in the denominator You just solve a simple equation for x and you obtain that x cannot be negative five because negative five plus five is zero So for the domain set builder notation we write that the domain is x Such that x cannot equal negative five That's what we mean by write the domain and set builder notation In part B. It's a similar experience where you have x squared plus one the denominator cannot equal zero So solving this little Equation so to speak except we have not equal to you have x squared cannot equal negative one Taking the square root of both sides You get that x cannot equal the square root of negative one is imaginary. It's a plus or minus I Now you don't care about imaginary restrictions. You only care about real numbers. That's what we're graphing in this section. So Don't care about the plus or minus I It's only real numbers we care about so the domain is actually all real numbers You can put any real number you want to into this function Now graphing using transformations. There's two parent functions. We'll be dealing with for rational functions There's one over x and then there's one over x squared These are the two parent functions for this section So the function one over x the y-axis serves as something called a vertical Asymptote we'll talk more about this in the next video But pretty much this is a vertical line that the graph will never ever cross The x-axis serves as something called a horizontal asymptote So in this case the graph will never cross the horizontal asymptote either So what happens is the intersection of this vertical asymptote and this horizontal asymptote these lines that this graph doesn't ever cross creates four regions There's a curve in the top right region that approaches the asymptote But never officially touches them, but they do get very very close And then the bottom left region is another curve that hugs those asymptotes gets really close, but never actually touches them That's for one over x Now one over x squared you still have a vertical asymptote as the y-axis You still have a horizontal asymptote as the x-axis Notice the asymptotes intersect at the origin zero zero Once again, this creates four regions where one over x squared is always a positive value for y Therefore this top right region we have a curve that hugs the asymptotes gets close to them But never actually touches them and we have a curve in the top left region Once again the curve gets close to the asymptotes, but never actually touches them So how do you graph these parent functions using transformations? Well when graphing using transformations, we will always start with the point zero zero But understand that this point is the intersection point of the asymptotes So in our next example we're going to graph using transformations and we'll give the domain and range So you have y equals one over x plus two followed by a minus four Now what you have to do is you have to look at okay x is in the denominator So this is a type of rational function So it's your job to first identify who is the parent It's either one over x or one over x squared the fact that you don't see any sort of quantity squared in the denominator of this fraction Suggest that our function our parent function will be one over x And now I'm just going to draw a rough sketch off to the side here about what one over x looks like you have a vertical asymptote And a horizontal asymptote the curves are in the top right region and The bottom left region So now we have a couple transformations happening here this plus two Since it's happening within the fraction with x is a horizontal shift. It's a shift left to This minus four after everything it's outside of the fraction. It's after the function. It indicates that we should go down four So remember we're going to start with zero zero first so This as a reminder Start with zero zero that is the intersection point of the asymptotes So you have a parent function. We're going to shift left to and down four so from zero zero I'm going to go left to Down four that's going to be where my asymptotes intersect So I need to draw my dashed vertical line to represent my vertical asymptote and Then a dashed horizontal line through that point that represents my horizontal asymptote So focus on the four regions created by the intersection of the asymptotes There's a top right top left bottom left bottom right The top right and bottom left region would be where we draw our curve so This is my top right region. Remember we hug the asymptotes, but we don't actually ever cross them top right and then bottom left So now to find the domain Notice that this graph is defined everywhere to the left of the vertical asymptote in terms of x and it's defined Everywhere to the right of the vertical asymptote in terms of left in terms of x so The only time that there's a restriction on the domain is when x is negative 2 You can look at the function algebraically and say yes negative 2 plus 2 will make the denominator 0 So my domain is pretty much going to be X such that x cannot equal negative 2 My range you're defined everywhere below the horizontal asymptote You're defined everywhere above the horizontal asymptote It's a y value of negative 4 where the function is not defined So the range would be y such that y cannot equal negative 4 That's one example of graphing using transformations and now one more So my function here is going to be y equals since you see quantity squared in the denominator. It's 1 over x squared So draw a quick sketch of what 1 over x squared would look like You have your asymptotes with curves in the top right and top left region Priority one is this negative sign out front that represents an x-axis reflection So let's think about what this x-axis reflection would do to my graph So consider your asymptotes which are on top of the axes in the case of the parent function If your curves are in the top right and top left region Then they will be reflected to the bottom left and bottom right region This is an x-axis reflection Lastly, we have a minus one that is with the x so we should shift right one And a plus two at the end is up to all right. So that being said we now want to Start with zero zero Always go right one up to That's going to be the new intersection point of my asymptotes vertical asymptote and Horizontal asymptote So I have my intersection point with the vertical asymptote going through it and the horizontal Asymptote going through it my curves belong in the bottom right and bottom left region. So we have bottom right region and bottom left region Remember you hug those asymptotes you get close to them, but you never actually pass through them so now To find the domain the set of all x values where the functions to find is to find everywhere to the left of the vertical Asymptote and to find everywhere to the right. It's not defined at x value of one So my domain would be x such that x cannot equal one And notice that's the value that causes your denominator to be zero the range you're only defined Below the horizontal asymptote you're defined for y values that are under negative under positive two You're not defined at positive two, but it's strictly below positive two So my range is why such that y is less than positive two So that's another graphing using transformations example and finding the domain and range and set builder notation Now I give you a type of rational function It's a rather strange one. That's already graphed and I want us to work together to find the domain the range and the asymptotes So notice the graph is defined to the left of the vertical asymptote x equals negative two to the right of the vertical asymptote x equals two and in between the two vertical asymptotes So the domain is x such that x cannot equal negative two x cannot equal to the range is why such that The first thing to observe is that below or less than negative two That's where the graph is defined these curves approach the horizontal asymptote negative two, but never actually cross it So the first part of your range is less than negative two But there's another part The range doesn't kick back on again until you get to a y value of exactly negative one You're defined at negative one and above So you would say add another component to the range of y is greater than or equal to negative one And that's your range and set builder notation your horizontal asymptotes would be x equals negative two That's the equation of the vertical line of the first vertical asymptote and x equals positive two lastly The horizontal asymptote has a y value of negative two So it's Y equals negative two. It's a horizontal line. It's y equals some number So that's an introduction to rational functions finding their domain and graphing them using transformations and leading to finding their domain and range Using the graph as well there. So thanks for watching