hello class and welcome to today's algebra lesson which is about piecewise and step functions by the end of today's lesson you will be able to graph piecewise functions and identify their domain and range and also graph step functions and identify their domain and range so we're going to start with the piecewise function a piecewise defined function is a graph that has different rules or different equations for the different intervals or different sections of its domain so essentially what it is saying is you can break the graph down into different pieces that follow different equations in this particular example that is shown down below here there are two different pieces of our piecewise function one moving to the left and one moving to the right so the first thing we're going to do is talk about how we can graph those so in order to graph those we are going to graph piecewise functions by using tables again so we're going to have our x y table the first function says that i've got x plus 3 for the x values that are less than or equal to 4. so i'm going to start my table at 4 and then because we want numbers that are less than or equal to my next two numbers are going to be 3 and then 2. so because it says less than i started at 4 and went down now that we have our values i would plug those into the equation for just the top part so i'd say 4 plus 1 which is going to give me an answer of 5 then i would say 3 plus 1 which is going to is going to give me an answer of 4 then i would say 2 plus 1 which is going to give me an answer of 3. so now i need to put those points onto the graph so i would go over to 1 2 3 4 to the right 1 2 three four five up and i would put my first point on my graph then i would go to three and four and then i would go to two and three so when you are connecting now you only have an arrow on the end where your points are trending so i started at four and then i went three two one so this graph is only moving and only covering the values that are moving to the left the ones that are smaller than four and because it is a linear function we want to make sure we have that arrow on the end then i move on to my second function or second part of my function and once again i'm going to start with four so my first value into this table is going to be 4. now because i've got a fraction in this piece i want numbers that can be divided by my denominator so i've got 4 and then i'm going to go to 6 and then i'm going to go to 8 because o's can all be divided by 2. so i'm now going to plug those values in and i'm going to say negative 3 halves times 4 which is going to give me negative 6 plus 6 giving me an answer of 0 then i would plug in 6 and negative 3 halves of 6 is going to give me negative 9 so then my value here is negative 3 when i add the 6 and then i do net 8 times negative 3 halves which is going to give me negative 12 plus 6 is negative 6. when i go to graph this second part notice this piece of your graph does not have the equal to part whichever part does not have the equal to part so in this case the second part your first dot when you put it on the graph so when i go over one two three four instead of having a filled in dot i'm going to leave that as an open dot then when i go to the next point 6 and negative 3 then i'm going to fill it in again and then i'm going to go over to my next point which is 8 and negative six and once again i can fill in that dot then i'm gonna go ahead and connect my points once again starting at my first dot and then moving through my second and third dot as i continue that trend and then adding my arrow back on to the end so when you are graphing piecewise functions you're going to have two separate tables both that start with the same x but both tables are then going to move in the opposite direction our last piece here is to find the domain and range remember your domain is your inputs or your x values and you can plug in any possible x value so the domain for these piecewise functions that we are going to be studying are always going to be all real numbers you can always plug in any x that you need to it's just a matter of whether the x will get plugged into the first part in this case the number smaller than or equal to four or rather we're getting plugged into the second part the numbers bigger than or equal to or just bigger than four for our range that's where the graph becomes important remember our range is our y values so as i'm going along this first line we're going up up up up up up up until we hit a top which is five then i drop down to my next line and then i'm going down down down down down so we do have a peak point of this graph which means that all of your y values all of the values going up and down are going to be less than or equal to 5 because 5 is the highest point that my graph can reach as we are moving go ahead and try making the table for just this first equation on your own so we have our first equation plugging in one two and three because we want the values that are bigger than one that gives us this line on the graph now go ahead and make the table and graph the line for the second part of this function so for this piece on your own so we plug in our values of this time 1 0 and negative 1 and we get this second line on our graph now go ahead and identify the domain and range of this particular function once again our domain is all real numbers i can plug in any value for x and then my range this time both lines are trending up and the smallest point that it gets to is negative three the other part of functions that we are going to be looking at is the step function this is a particular kind of piecewise defined function so a specific group that we are breaking it down into that consists of constant pieces basically it means your answers can only be one defined set of values the particular one that we are going to look at is called the greatest integer function so this is a type of step function where the answers are the greatest value or the greatest integer less than or equal to x so even though we're looking for the greatest it says less than or equal to x so we are rounding down we know that we are going to do this rounding when you see the x values in these fancy brackets and once again it means all values since we want less than or equal to are going to be rounded down there's a couple different scenarios we're going to end up with in this particular case the first is if our values all lie inside of those fancy brackets when you are doing a step function and you are making your table because once again we graph our piecewise functions by making a table we cannot just plug in whole numbers so i can't just go negative 1 0 1 2 etc i need to plug in values that are going to allow me to do some rounding so i'm going to plug in y i'm going to start with negative 1 then i'm going to stick in negative 0.5 i'm also going to stick in negative 0.25 so we want a couple values couple decimal values in between negative 1 and 0. we're going to use a couple of decimal values that are between 0 and 1 and then we are going to have our value of 1. so we need some decimal values in between these values here i'm actually going to use a few more just so that we can really see the point and i'm going to add in negative or extruding positive 1.25 and positive 1.5 as well now normally we just have our x and our y's but because of these brackets we actually are going to have a third column because this problem involves rounding because our value of x minus 4 is on the inside my next part is just going to be to solve the x minus 4 so to figure out what the inside values are actually going to be so when i solve those i have negative 1 minus 4 which is negative 5 negative 0.5 minus 4 which is negative 4.5 and then negative 4.25 here and then we've got negative 4 and then we've got negative 3.5 then we've got negative 3.25 then we've got negative 3 negative 2.75 and negative 5. so as i do all of that subtracting i get these decimal values the last column then of this table is going to be to actually do the rounding so putting that fancy bracket in there to say you are now doing the rounding as you do the rounding we want to round down to the nearest hole so these values that have no decimals stay as whole numbers then i look at my decimal part i want the number that is below negative 4.5 with no decimals so that's going to be negative 5. i want the number that is below negative 4.25 with no decimals so that's negative 5. i want the number that's below negative 3 and a half which is negative 4 and then negative 4 again and as i look at this the number below negative 2.75 or below negative 2.25 is negative 2. now when i put this on the graph that means at negative 1 negative 0.5 and negative 0.25 so all of these values in here my answer is going to be negative 1 2 3 4 5. so at negative 1 at negative 0.5 at negative 2.5 all of those have an answer of 5. when you are graphing these step functions this particular greatest integer function you're always going to have a filled in dot on the left a flat line and then an open dot on the right when i go to my next one 0 to 0.5 to 0.75 that's going to give me all of the values that are negative four and then i've got for the next one all of the values that are negative three that only builds three steps of my step function but that is enough for me to keep going and build in the rest of the step function so in your table you need enough to initially start filling in to figure out what the actual rounding values are going to be but once we actually start putting them on the graph you don't actually have to solve for all of these pieces that i have been adding as i continue to talk here all of these pieces would be the correct rounding if i added on to my table but they do not actually needed to be added on there notice every step we've got one filled in dot on that particular vertical line and one open dot on that vertical line so it ends up looking like a staircase which is why they call it a step function when you fill in your graph for the step function you should be going from however far down you can get to however far to the right or to the left depending on which way your stairs are going that we can fit i'm going to do one more example of these ones with you together this is where we have our value on the outside so it could be in any of these step functions adding subtracting multiplying or dividing this first one so i happen to have subtracting and it was on the inside this next one has multiplication and it's on the outside so when you are building a table and that factor is on the outside once again because it's a step function i need to make a bigger table so i'm going to start again with negative 1 throwing in a couple of those decimals so negative half negative 0.25 then i get to 0 and then i do my decimals of 0.5 and 0.75 then i'm up to 1 and i'm going to go with 1.25 and 1.5 so then i'm going to go ahead and do the interior piece first so in this case it's asking you to do the rounding first because the rounding is what is technically inside the parentheses so that means what i need to do is round the biggest number below my integer so once again my integers themselves stay the same the number below negative point five is going to be negative one the number below negative point two five is going to be negative one the number below point five is zero the number below point five five is 0. the number below 1.25 is 1 and the number below 1.5 is also 1. so in this case we did our rounding piece first and then the last piece of this table would be the negative and then multiplying by our rounded values so when i do this one i've got negative 3 times negative 1 which is positive 3 negative 3 times negative 1 which is positive 3 negative 3 times negative 1 which is positive 3 negative 3 times 0 is 0 0 and 0 and then negative 3 times 1 is negative 3 negative 3 and negative 3. once again then we take our values and we are filling them in so from negative 1 to 0 i'm going to have a value of 1 2 3. my next step then is to go to 0 and 0 and that is my next step in my function and then i go over to 1 and i go down to 1 2 3. now notice this time there's a gap in our stairs so when i go ahead and fill these in i'm going to go down again one two three to put in my next set of stairs and then once again one two three down to put in my next set of stairs i can't just go down one space and fill in a set of stairs so that's all the farther i can go as i'm going down on the opposite side i would go up one two three spaces and put in another step up one two three spaces and put in another step and again that's all the higher you can go so when we have this multiplication your stairs are going to get stretched out a little further than they were before the piece now that we need to look at with either of these functions is our domain and range once again our domain for these piecewise functions is going to be all real numbers i can plug in any value in for the x spot fractions decimals square roots whatever it is you can plug in that value for x the second part here is we have as i look at my steps i have all of the values here which would be at negative 10 negative 9 negative 8 negative 7 negative 6 negative 5 so all of those values are all integers because there are no fractions or decimals included in the final answers of our steps when i come to this next function once again our domain is going to be all real numbers because you can plug in any value for x my range this time the steps get bigger i go from 3 or excuse me from 9 to 6 to 3 to 0 to negative 3 to negative 6 to negative 9. so we have those bigger jumps of our steps in order to indicate that we would say all integers so we're still talking about values that have no decimals but then we would say multiples so talking about multiplication of 3 because every time we jump we're jumping three spaces if it had been multiplying by negative seven every time you jump it would have been seven spaces and you would have said all integers multiples of seven so whatever your number is in your equation that's going to help you figure out what your value is as you are doing your multiples go ahead and make the table for this one and the graph on your own so we fill in our table we do our rounding first in this problem because there is nothing else inside the parentheses or those special parentheses excuse me and then we do our adding afterwards so we add our values on and get 0 1 and 2 filling in that whole graph now i want you to go through and name the domain and range of this particular function once again your domain is all real numbers and your range is going to be all of the integers if you have questions about this or anything else from the lesson please feel free to reach out and let me know