hi everyone this is Mrs J and today we're going to learn about functions continuity linearity intercepts and symmetry so we're really starting to talk about some of the key features of functions I have a quick warm-up for you guys so please pause the video and give it a try all right go ahead and check your work here this is just to review a couple basic skills from algebra one so the first one is substituting values and making sure you're following order of operations to help you evaluate the second one is just solving for different variables so isolating that variable within an equation and whether you write your answer like this or this they're completely equivalent so of course when we start talking about functions the first thing we really need to talk about is what exactly is a function what is that definition so a function is the special type of relation where every input has only one output that is the essential definition of a function which essentially means every x value has only one y value so this is important to us because you know functions actually model real life scenarios so this is saying that every time we plug in some input X every single person in the entire world would get the same y value they would get only one y value or only one output so again this is the definition of a function and it's super important to keep this in mind throughout the entire years we start learning about different types of functions so there are two ways we can classify functions we can call a function a one-to-one function where in this case every output also only has one input so again we already know that it's a function but a one-to-one function again every output has only one input and or we could say that it's a mini to one function which is a function where some outputs have multiple inputs so just some different representations of some number relation so in this relation here we can see it is a function because every input has exactly one output compared to this relation here where the input of 9 has two different outputs which makes it not a function but this one again you could see that each input has one output and every output only has one input but then if you look down here at this relation it is still a function every input gives you exactly one output however you can see that this output of 4 has two different inputs which again is fine it still makes it a function but we call this a many to one function because there are multiple inputs that give you the same output so how can we tell graphically if something is a function so for this the way we test to see if every input has one output is with the vertical line test so the vertical line test essentially says that you should be able to draw a vertical line at any point on your graph like this and that vertical line should only cross your graph at one point so you can see on this first example no matter where I draw a vertical line it only Touches at one point so that means that every x value has only one y value which means it is a function so yes now if you look at this next example same thing wherever I draw a vertical line My Graph only crosses that vertical line once so every input has exactly one output but if you look at this third graph just look at right here and this happens anything to the right of negative five if I draw a vertical line This input say of negative 2 has two different outputs so this because it fails the vertical line test is not a function now once we know if it's a function how can we tell if it's a one to one function so for this we're actually going to be using the horizontal line test so again we've already we're only using this if we've already determined that is the function so the horizontal line test we would draw a horizontal line at any point like here here or here and if it only crosses once then we would call it a one to one function so this first graph is a one-to-one function but if you look at our second graph which again we already know is a function if you draw a vertical line let's say here we can see that it crosses twice so that means that this output of negative 4 has two different inputs again it is still a function but we would classify this as a mini to one function and again for this third one it's not a function so we wouldn't classify this as either because we're only going to do that once we know it's a function um so now we're going to talk about a different way we can characterize a function and those are with either the word discrete or continuous or maybe we would say that it's neither so a continuous function is simply a function if you think about it if you can graph it without picking up your pencil so it's graphed with an unbroken line or curve we would call it continuous and then a discrete function is a discontinuous function that's just simply made up of unconnected points and then if it's kind of neither just points or a connected curve we would call it neither continuous nor discrete so if you look at let's start over here on this downward facing Parabola I see that I could draw this entire graph without picking up my pencil so we would call this a continuous function and if you look at this graph in the middle it's just a series of unconnected points so we would call this a discrete function and you'll notice that at no point are there any connected coordinates it's just unconnected points now if you look at this first graph right here there are portions that are continuous but then there's a jump and then it's continuous again so since it does have that jump it is discontinuous and it is also not discrete so this one we would classify as neither and again this is just a different way that we can characterize different types of functions and now we're going to talk about a really important concept that will keep returning to over and over throughout the year and that's domain and range so hopefully these these words sound familiar to you we know that domain is the set of X values or the input that is defined for any function and the range is the set of Y values or the output that is designed for any function now here we're going to talk about different types of notation I'd say as you progress through higher levels of math we're going to kind of gravitate towards interval notation but it is important to know how to read and write out also algebraic and set builder notation so we're going to practice all three so you can see here on our number lines we have four different relations of numbers and we're going to talk about how to write these different scenarios using all three notations so here one thing to notice is that here we have an open coordinate at negative two anytime you see an open coordinate or this open circle it means that this point is not included as part of our domain or range here we're going to do all domains these are X values so it is important that we do not include negative 2 but everything to the right is included and that is denoted with this little arrow so it means that everything that is larger than negative 2 is included so when you're using interval notation we always start with the low number on the left and the highest number on the right so you can see that our lowest number is negative two and since it's not included we're going to be using a parenthesis so this is called an Open Bracket so we use this when the point is not or the value is not included and then we put a comma and then to the right we include our highest value now if it's just continually or continuing infinitely to the right our highest value is considered to be infinity and infinity always goes with a parenthesis or an Open Bracket like this it's never technically included and I like to think of it as well because it's not technically a number so we can't technically include it so this is how I would describe this domain in interval notation low to high and we can see that negative 2 is not included now algebraically we're essentially just going to be using inequality symbols so here we're describing domain so it's X and we can see that's everything larger than negative 2. and then set builder notation is really a lot like algebraic notation but with a little bit extra kind of fancy stuff going on so here we're actually going to start with our curved bracket or a brace and then we say what variable we're working with so X and then we do this vertical line it's red X such that and then we essentially just do our algebraic notation X is greater than negative 2. so very similar to algebraic with our braces and defining the variable we're using in front okay let's look at this next one so here we can see that it's everything below and including or positive three so with interval notation again it's always low to high since it is continuing infinitely to the left our lowest value is negative Infinity never included our highest value is 3 and this one is included because it is a closed Circle so we're going to use this bracket so this is a closed bracket showing that 3 is part of our domain for this one it's everything less than or equal to three and then for set builder we'll say x such that X is less than or equal to 3 with our braces now if you look at this next example we actually have a lower and upper boundary that are both defined by numbers so our lower boundary is negative four and because that is included it's a closed coordinate we're going to do a bracket and our upper boundary is positive one however it's an open coordinate so we're going to use a parenthesis to denote that is not part of our domain now when we have something like this where it's between two numbers we're actually going to put our X and our inequalities in the middle so we'll still start with our lowest value and then it's going to be greater than or equal to negative 4 but then less than positive 1. so this kind of shows that it's between negative 4 and 1. negative 4 is included one is not then we'll do just our extra braces X such that negative 4 is less than or equal to X is less than one and then for this last one you could see that it's just continuing infinitely to the right and left which means it is all real numbers so with interval notation we'll just always say that's negative Infinity to positive Infinity um when it is all real numbers with algebraic notation we're actually going to use the all real numbers symbol so we could say x equals all real numbers which is that capital r with the extra vertical line and then for set Builders we'll get a little bit fancier we'll say x such that X is represented by all real numbers so you can see for this one I'm kind of using this sideways kind of e shape to denote that it is all real numbers so these are the three different notations we'll be working with and we kind of covered all the different scenarios where if points aren't included or not and what to do as the values move infinitely to the right or left all right so now we're going to look at some different graphs and see if we can Define our domain and range we'll use interval and set builder notation so let's start with our domain for this graph so here we can see that with these arrows it's showing that it is going up forever to the left and right but you can also see that it is going over a little bit to the right and left so although it's going up much faster it will keep extending to the left and right infinitely so here our domain is all real numbers so negative Infinity to positive Infinity in interval notation and then we could say x such that X is all real numbers for a set builder notation now for our range we can see that our range does have a restriction it does hit a low point at negative four and then it's everything above so this will go from negative 4 to infinity and because our graph goes through the coordinate 0 negative 4 negative 4 is part of our range or if we were using set Builders it'd be y such that Y is greater than or equal to negative 4. so make sure you're using the correct variable when you are using that set Builder's notation all right let's try another one together so kind of similar to the last example you could see that these arrows denote that it is going down forever but it's also going slightly to the right and left forever so for our X values it will eventually hit all real numbers to the left and right so negative Infinity to positive infinity or X such that X as represented by all real numbers and then for our range which again is the Y values we can see that it's everything below and including positive five so remember for interval notation we always go low to high so negative Infinity to 5 and I used a bracket because 5 is part of our range or we could say why such that Y is less than or equal to 5. and notice that again when we write this vertical line we read it as such that y such that all right let's try one more together so here we you see that we actually have what's called a piecewise function so it's kind of split up into different pieces and some of our coordinates you'll notice are open and some are closed so let's start with our domain so as long as the x value exists somewhere on our graph we will include it in our domain for example I know that this is an open coordinate here at x equals negative two but since it's a closed coordinate here negative 2 is part of our domain somewhere so we will include it same here for positive 4. I know this is open but here it is closed so it exists somewhere now if you look at negative 6 this is a closed coordinate so negative 6 will be our lowest x value but if you look to the right our highest x value is at eight but that coordinate is open so 8 will not be included in our domain for set Builders we can say X such that negative 6 is less than or equal to X Which is less than 8. so make sure you're not including eight in your domain now for our range let's take a look so again as long as the Y value exists somewhere we will include it so here around y equals 5 I know it's not included here but it is included somewhere else here at six it's not included here but it exists somewhere so really our range is going to be everywhere between I'll just estimate that this is negative seven all the way up to 9. so again as long as it exists somewhere we will include it so our low value looks to be negative seven our high value looks to be positive nine and for set Builders we'll say y such that negative 7 is less than or equal to Y is less than or equal to nine so make sure both negative seven and nine are included in your range all right so now I have one for you to try on your own so please pause your video and then you can hit play when you're ready to check all right go ahead and check here I know that there's all these different pieces to this graph and there's some open coordinates but really you can see that everything above zero is included for our range and zero is included it's included somewhere and you can see that our graph actually goes to the right and left infinitely which is why our domain is actually all real numbers every x value is included somewhere on our graph all right so the next thing we're going to talk about is what's called linearity so a linear function which is one of the first types of functions we're going to be working with is a function which has no variable in which no variable is raised to a power other than one okay so again no variable is raised to a power other than one so there shouldn't be anything being squared there shouldn't be any square roots or anything like that all linear functions can be written in the form of y equals MX plus b where m and b are real numbers so that's one way to check to see if something is linear if you can rewrite it in this form graphically we know that something an equation is linear if the graph makes a straight line or we could say has a constant slope okay so let's look at these three equations and see if we can determine whether or not they are linear so this first equation I want to remind you that f of x is completely interchangeable with Y we can actually just divide each of our terms by three and you can see that it can be written in this form meaning that it is linear you can also see that both X and Y are just raised to the power of 1 which another way to determine that it is linear for that first equation now if you look at this next equation right off the bat you'll see that X is being cubed being raised to the power of 3 so we can see that this is a non-linear function and then for this last equation I know that maybe you're thinking oh this is not linear because I don't even see an X but remember m and b can be any real number and 0 is a real number so if you just have y equals 9 it actually just makes a horizontal line at the value y equals 9 and this is considered to be a linear function it has a specific name technically this is called a constant function which would be a type of linear function but this does make a straight line the slope is just zero and again this this will be considered a linear function another really important characteristic of functions will be The Intercept so both the X and the y-intercepts which is just the uh the x-intercept is just the x coordinate where the graph crosses the x-axis again it is possible for functions to have more than one x-intercept not linear functions but other types of functions can and then the y-intercept is simply the y coordinate of a point where the graph crosses the y-axis and to maintain the property of functions functions should never have more than one y-intercept or by definition they wouldn't be a function anymore so here we're going to practice finding our x and y intercepts algebraically so to find let's start with our x-intercept if our x-intercept is where the graph crosses our x-axis that is always going to occur when y equals zero so Define an x-intercept we set y equal to zero and we see what the x value is so I'm going to substitute y was Zero then I'm just going to solve for x so we can see that X is equal to negative 3. that's our x-intercept and then we could also we could write it as a coordinate like this so it'll be negative 3 0. and our y-intercept well if it's where it crosses the y-axis that will always occur when X is zero so we're going to set x equal to zero so we'll substitute into our equation and then we're just going to solve for y so y equals 9 over 2 so this would be our y-intercept so 0 9 over 2 would be the full coordinate and you will notice that kind of throughout the year I will be using mostly fractions rather than decimals which I would also like you to do always make sure they're simplified and notice here we are going to be using improper fractions so we will just keep this as 9 over 2 rather than 4 and 1 half so please get in the habit of being comfortable with improper fractions and just always make sure they are simplified all right the next characteristic we're going to talk about is symmetry so some graphs or some functions do display different types of symmetry but it's also very possible for a graph to show no symmetry at all so it can be neither of these things but there are two main types of symmetry we are going to be looking for the first one is called line symmetry and that's when your graph is reflected over a vertical line so that the two halves are equal so if you look in this at this Parabola which is a quadratic function if you draw this vertical line here you can see that the graph to the right is completely identical to what's to the left of that vertical line or if you folded your paper on this vertical line the grass would meet up so this is called line symmetry and we could even Define the line that it is symmetrical over and then if you look at this next graph this graph displays what's called Point symmetry so that's when a graph is rotated 180 degrees which is half of a circle about a point to map itself onto our maps onto itself so you you can see here that if you took this half of our graph and we kind of rotated it about in this case our point is the origin if we rotate it about the origin it would map perfectly onto this half of the graph another way you can kind of check for this graphically if you pick any point on your graph and you draw a line through that point and your point of rotation which again in this graph is the origin you can see that if you continued that line the other point that it goes through would be the opposite of those y values and is the same distance to your point of rotation so we can kind of use that to help us determine where that point of rotation would be so these are the two types of symmetry we're going to be looking for and again it's very possible for a graph to not display either of these types of symmetry and that's fine as well okay so here we're looking at three different graphs and trying to identify if a type of symmetry exists so here we actually have a linear equation and actually all linear equations actually um are they display Point Symmetry and actually the point of rotation could be any point on the graph so all of this graph any linear equation does have Point symmetry and the point of rotation could be I'll say about any point on the graph and again you know I could take this point and you could see maybe if I took uh this point right here and I rotated it 180 degrees it would land right here and we could take any point on the graph as the point of rotation and pick any point to the left or right and rotate it 180 degrees and it should still fall on your line now if you look at example B this portion of our graph looks like it does have that rotational symmetry however you can see that then we have this portion of the graph so there is no rotational symmetry and we could also see that there's no line symmetry because it's not reflected exactly left to right so this one would be neither and then if you look at this third example here we can see that there is a vertical line that kind of reflects our graph that vertical line would be right here um I know the scaling on this graph is a little bit strange but each tick mark should be one I think some of these points just got kind of shifted over so we would call this line symmetry and we would say about and we're going to Define this vertical line and this is x equals negative three so it's about the line x equals negative 3 and you could again if you folded your paper on this line we would see that both halves of our graph would match perfectly and there you have it the last thing we're going to talk about today is yet another way that we can describe some functions and that's why either classifying them as even functions or odd functions now it's also very possible for functions to be neither even nor odd but if you can classify them as we kind of move forward throughout the year you'll see why this will be useful to us as we learn to graph and we learn about different features of our functions so first an even function there is an algebraic test for this which is kind of the mathematical definition of an even function so here's the algebraic test it says f of negative x equals f of x so what does this mean well you can see what it means is if you plug in the opposite x value so if you substitute the opposite x value and notice I'm not using the word negative I'm just using opposite so we have X and negative X which would be opposites but plugging in or substituting the opposite x value actually gives you the same result as if you plugged in f of x so it results in the same y value so if you substitute the opposite x value you get the same y value that would be like if you plugged in negative 5 and the output was positive 2 and if you plugged in positive 5 and the output was also positive 2 we could say that that is an even function as long as that holds true for any x value so opposite X values give you the same y value graphically a graph that is or a function that is considered to be even will always be reflected over the y-axis so if you notice that it has line symmetry about the light the y-axis this would be considered an even function so you could either look at the graph and determine or you can you can do the algebraic test now for an odd function the algebraic test or the mathematical definition is that f of negative X equals negative f of x so what in the world does this mean this means if you substitute the opposite x value again if you substitute the opposite x value you get or the output is the opposite y value so plugging in the opposite x value gives you the opposite y value you get the opposite y value so graphically you can tell if a function is odd if it has Point symmetry about the origin meaning it is rotated 180 degrees about the origin kind of like that first kind of cubic function that we looked at so it is going to be useful for us to be able to either look at an equation and do an algebraic test or look at a picture of a graph and determine graphically if we have an even or odd function or maybe it is neither so we're going to give a couple of these a try so here we're going to do the algebraic test for these so anytime you're trying to test what we do is we plug in Negative X or the opposite X and then we're actually going to evaluate this in our function and see what happens so here we'd have negative 2 times negative x to the power of 5 plus 3 times negative x to the power of 3 minus 7 times negative X now if we simplify this we know that if you have a negative value being raised to an odd power the result is a negative so this will actually give you negative x to the fifth so if you multiply it by negative 2 this actually becomes positive 2x to the fifth same idea here if you have Negative X cubed that is negative 1 times x cubed so that turns this entire term into negative 3x cubed and here we know negative 7 times negative X gives us positive X so what you'll notice is when we plugged in the opposite x value you'll notice that our entire oops sorry I forgot an X our entire function was made opposite so really what's happening here is that f of negative X gave us negative f of x plugging in the opposite x value gave us a completely opposite y value so again every single term in my function was made opposite meaning it changed from either negative to positive or from positive to negative so because that is happening this function is classified as odd now if you look at Part B let's do that same algebraic test we have 0.5 times negative x squared plus 11. so let's simplify that we know that if we square a negative it will always result in a positive so this first term actually just stays exactly the same again squaring it just essentially negates that negative and then 11 is a constant right there's no variable it will never change 11 will always be 11 no matter what we're substituting into our function so when we substituted Nate the opposite x value we actually got the exact same y value so here this is f of negative X is equal to just the original function f of x and this is the definition of an even function now if you look at part C let's do our algebraic test so I'm substituting Negative X this first term we're cubing it so it turns the entire term negative here this will become minus 3x but our constant will never change so here our first term became opposite our second term became opposite but our third term did not so here we would not say so f of negative X is not equal to negative f of x not everything became opposite um but we would also say that it is not equal to the original function it is not exactly the same so if it's not exactly the same and it's not exactly opposite this is a case where we would say that it is neither even nor odd so there is kind of a little cheat that you could do when you're looking at an equation if you're looking at a function and you notice that all of the variables every single term has a variable with an odd exponent that function will be odd and if all of the terms have a variable with an even exponent or no variable meaning it's a constant if everything is even or constant the function is even now if you have a mixture of even and odd exponents or odd exponents with a constant that's when it's going to be neither so that's kind of the shortcut but we would still want to to be comfortable with this algebraic substitution to determine if something is even or odd all right and that is all for today's video thank you so much for watching