 Hello, so today we're going to be working on combining functions and the first thing that I want you to notice here is a phrase that I put up here at the top of this worksheet called like terms. I'm going to be referring to that quite often today and in the future for sure and like terms is when the variables and their exponents are exactly the same so they have the same degree but the coefficients are often different. Now the coefficients can be the same, that's kind of a fluke but not a big deal there. Also I took right out of our textbook here some function notation. So addition you will sometimes see as f of x plus g of x or you will also see it as f plus g of x. So remember these parentheses do not mean multiplication, they mean of x just like with functions we talked about in the last chapter. So that's the minus in here and with addition back to this one all you have to do is combine like terms. Subtraction you do want to remember that the negative sign here is going to have to go to your g of x function, whichever that one is. Multiplication, we're going to be dealing more with this shortly but basically you're just going to multiply the two functions together which means distributing and then division, this one actually gets written three different ways, gets written as a fraction with a division sign and then also just one function on top of the other function. So when you combine the division here, you notice down here at the bottom, they do make a case that you cannot divide by zero so you do want to remember that. Okay so for our examples here today we're going to work with f of x is 3x plus 8 and g of x is 4x minus 10. So the first thing we want to do here is an addition problem so we want to add these two functions together. So you can write this one of two ways, I'm going to write it just one next to the other one so 3x plus 8 plus 4x minus 10. If you prefer you can always put one on top of the other one to help you see what goes on there but no totally your choice. Okay so now I'm going to look for like terms so we've got the 3x and we've got the 4x so when you combine those together you get a 7x and then we've got the 8 here and the negative 10 here is our constants. So when you combine those together you get a minus 2. So whenever you add functions, subtract functions, multiply functions, you will end up with a function unless there's a number in here instead of a variable. Okay when we go to do f of x minus g of x is pretty much the same thing except and this is a big exception. You need to make sure to throw the second one in parentheses, 4x minus 10. So that way you remember to distribute your negative sign to both pieces in that second function. Okay or you can just remember to subtract everything but I prefer to distribute. So I'll write this as 3x plus 8 then that will end up giving us a minus 4x but then a negative of a negative is going to give us a positive 10. Again go through and combine your like terms so we have a 3x and a negative 4x this time. So you want to combine those together and get a negative x or negative 1x if you prefer to write it that way and then here with this one we've got a positive 8 again but this time we have a positive 10 so when you combine those together you end up getting a plus 18 so that would be your final answer there. Okay next multiplication again I like to throw parentheses in here so I'm going to write parentheses 3x plus 8 parentheses 4x minus 10. So everyone's favorite this is a binomial times a binomial and you guys probably all remember foil first order and last I prefer just to distribute myself. So I'm going to use a couple different colors here so you can see so the 3x needs to be multiplied by the 4x as well as by the negative 10 and then here we need to multiply the 8 by the 4x and then the 8 by the negative 10. So when you do 2 pieces by 2 pieces you're going to end up with 4 pieces total and then we'll combine like terms. So when I multiply 3x by 4x that gives me a 12x squared. I multiply 3x by my negative 10 that gives me a negative 30x. Then I multiply the positive 8 by the 4x that gives me a positive 32x and then a positive 8 times a negative 10 gives me a negative 80. So then combine our like terms I only see 2 like terms and those are my x's in the middle here. So when I combine those two together I get a final answer of 12x squared plus 2x minus 80 and that is, oopsie, our final answer. Let me erase that and make that look a little bit nicer there, there we go. My circle didn't work out so well, there's a final answer. Okay and last but certainly not least we have f of x over g of x and all you have to do is just write out your functions. So this will be 3x plus 8 over plus I said 4x minus 10 and that is your answer. Now this is actually called a rational function and we'll deal with those later on in the semester so that way we know we can't divide by 0 so we can figure out what value of x will do that, etc, etc. Okay also back here with part c we're going to be doing more of these problems shortly and I'm going to show you a more visual method to do it as well because I like the visuals. Okay so this problem is similar but yet very different. You notice we're not actually given functions for f and g but we're given values here that come off of the table. So the first problem here is an addition problem. It wants me to add f of 2 and g of 2. So that means my input here my x is 2 so I need to go to my x is 2 table here and then use these values off of my table. So that means I'm going to do f of 2 so f of 2 is 15 plus g of 2 which is 5. So when I add those two values together that gives me a grand total of 20. Next one I'm going to do f of 1 times g of 1 so this one's a multiplication problem. Again you'll notice that the inputs are the same. My inputs this time though are 1 so that means I need to go to my x equals 1 and I'm going to use this column of my table. So I'm going to multiply f of 1 which is 10 times g of 1 which is 3. Multiply those two values together you get a grand total of 30. Next problem is a subtraction problem and again our input this time is 4 and you notice it's the same for both. It doesn't have to be but that's typically going to be the way you're going to see them. And since it is subtraction the order of these really does matter. So g of 4 is 9 so I'm going to do that one first minus f of 4 which is 25. So when we do 9 minus 25 I believe that gives us a total of negative 6. Then the last problem here is a division problem and this time our input values are 2. So we're going to have to go back to our 2 column and pick those values out. And f of 2 is in the numerator so that's going to be 15 divided by g of 2 is in the denominator that's 5 which is not 0 so this division is possible. And when we divide 15 by 5 we get a grand total of 3. Okay so these problems look a little bit different but they run pretty much the same way. You just have to know which column you're picking information off of the table and go from there. Okay the last problem this one's very similar to the table problem except this one you're given graphs. So you have to know how to read the graphs in order to be able to get our outputs. So you'll notice that the input for this one is 4 for both of these problems. So that means I need to go over here to x equals 4 and I'm going to pick off my y value. So on f my y value at 4 is 0. So that's going to be 0 plus. My g value at 4 however is looks like negative 7. So that's my g value here so that'll be 0 plus negative 7. So that gives us a total of negative 7. We're going to do the next one exactly the same except we're going to be using a different input value. This time our input value is 2 so when I go to my f function at 2 it is negative 2. This is a multiplication problem. My g value at 2 is looks like negative 5. And you notice I'm just eyeballing these they might not be perfect but it's close enough. So now when you multiply negative 2 by negative 5 that gives you a grand total of 10 and it's positive. Okay so practice some more of these look through your book look through your web design problems and hopefully you'll find everything that you need here to help you out with this stuff. Thank you very much for listening. Have a good one.