evaluating functions worksheet
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hello welcome back to algebra the title of this lesson is called evaluating functions part one in the last lesson we talked about the concept of a function in algebra which youll be using functions pretty much forever so just kind of get used to the idea but its a machine that takes input values in X and then it calculates it it gives you the output values that we call f of X so this lesson is all about doing those practice calculations and we call it by the way evaluating the function is what we call it when we take that input and we calculate the output we call it evaluating the function so if you had a function f of X is equal to a very simple function X plus 2 first of all what kind of function is this you should recognize they recognize this as a line em X plus B so you know its a line and we say we want to evaluate the function at X is equal to zero and evaluate the function at X is equal to negative one so when the value that we put in the middle here this tells me the value of x that I want to evaluate the function at now I want to do it two different times so the way you write it mathematically is you break the f you open the parenthesis and you take the value that youre trying to evaluate and then you substitute it in for X so zero plus two is you all know its two so in your paper you just say F of zero is equal to two this is what you write down now you have to be careful because up until now weve been using parentheses to mean multiplication but in this case when were writing functions down it never means multiplication it means the function f evaluated at x is equal to zero gives the value two thats what this means so it does not mean F times zero and by the way the letter F is almost never used for anything in math other than functions so when you see f you always know its a function now do the other one here we do exactly the same thing f
evaluated at X is equal to negative one means we stick negative one in here and we get what do we get one negative one plus two gives us positive one so we say F evaluated at negative one gives us one we circle those answers all right so were just going to continue working down the way here this is mostly an exercise in substitution but were kind of getting you used to the idea of what a function is you also very commonly will see the function G to represent the letter G to represent a function basically it doesnt matter what letter of the alphabet you use but typically youll see F youll see G in physics sometimes youll see U and V but almost always in algebra its going to be F almost always some times G of X so just just kind of get used to that you might see G of X representing a function that were naming G so this function of X is 1 minus 2 times whatever X is and lets say we want to evaluate it at X is equal to negative 2 and we want to evaluate it at X is equal to positive 3 so how do we evaluate that we take these values in and we stick them into the function G evaluated at negative 2 means that we take and then we stick it right into the value of x here so we have 1 then we have a minus sign here but this multiplication is actually negative 4 minus and this multiplication is a negative 4 or you could write it as 1 plus 4 however you want to do it its going to end up being 1 plus 4 and so youre gonna end up with 5 for the answer there and so what youre gonna get is G evaluated at negative 2 is equal to 5 and this is what you want to circle on your test the function G evaluated at X is equal to negative 2 gives you this value all right so the next one when X is equal to 3 means we take G we evaluate it at the position or X equal to 3 and we say its 1 minus 2 times 3 in here
so its 1 minus 6 and 1 minus 6 as you know is negative 5 so G evaluated at X is equal to 3 is negative 5 and thats the final answer for that one and again were just going to continue cruising along and doing different functions getting slightly more complex each time lets say we have the function of X as being the absolute value of 2 minus X so now we have an absolute value in the mix and want to evaluate it when X is equal to 0 and also when X is equal to negative 2 so well switch colors a little bit here and we will say F evaluated when X is equal to 0 is equal to 2 – zero with the absolute value around it so it means its the absolute value of two which as you know is just 2 so f evaluated at 0 is 2 thats the function evaluated there as well and then finally we will take the last one and we will say F evaluated at X is negative 2 is absolute value of 2 – now be careful its – but X is negative so you should wrap it in parenthesis just to make sure you dont make any mistakes 2 – whatever X is which is negative so wrap it up like that and then youll have 2 and this negative times negative gives you positive which equals absolute value of 4 which equals 4 so you write your final answer is a negative F of negative 2 is 4 all right so again were not doing anything too crazy were just kind of getting practice with evaluating functions so what if you had the function H which is pretty unusual to use H for a function name but you can you can use anything you want H is a function of X and thats going to be equal to 10 divided by x squared + 1 10 divided by x squared plus 1 so complicated looking function right you would definitely have to plot points to figure out what this is going to look like lets evaluate it at X is equal to 0 and also an X is equal to 1 all right so for the first one we have
H evaluated at 0 here we just plug the value in 10 over 0 squared plus 1 in the bottom so what we have is 10 over this is just going to be 0 this is going to be 1 which means youre going to get 10 is the answer so H evaluated at 0 is 10 thats the final answer for this and then well evaluate now at X is equal to 1 so H evaluated at 1 is 10 over X whoops not X we have to substitute in for 1 now 1 squared plus 1 so well have 10 over this is going to give you 1 so 1 squared is 1 1 plus 1 is 2 which is going to give you 5 so H evaluated at Y is five all right I only have one more just to give you a little practice what if your function looked like getting a little more complex now f of X is equal to 1 minus X on the top divided by X cubed on the bottom and we wanted to evaluate this at X is equal to 2 and again we want to evaluate it at X is equal to negative 1 so again you have to go through the mechanics of the algebra but basically its the same thing you stick the value of of x in two locations notice in all of these previous ones we had an X here we had an X here and so the other ones we only have one value of x here we have an x value in two different locations but its the same thing you just stick this value of x in both locations so what we have is f evaluated at 2 is 1 minus 2 on the top because were sticking two in there and then its 2 cubed on the bottom now what is 1 minus 2 its negative 1 now what is 2 cubed thats 2 times 2 times 2 so 2 times 2 is 4 4 times 2 is 8 so you have 8 so what you have is negative 1/8 so you write it as f evaluated at 2 is negative 1/8 and its perfectly fine to get fractions for answers when youre evaluating functions now for our
or I should say our our our next one here were going to evaluate at X is equal to negative 1 you say f evaluated at negative 1 is 1 minus now here you have to be careful 1 minus but X is negative so you should wrap it in parentheses to make sure you dont make any mistakes 1 minus this value of negative 1 and the same thing on the bottom its negative 1 cubed you should wrap it in parentheses to make sure you dont make any sign errors because if you dont put the parentheses here you might forget to cube the negative sign as well now on the top what you have is you have the 1 minus the negative 1 so you have double negatives here so it becomes a plus so its 2 and on the bottom Im going to help you out and write it like this this is negative 1 times negative 1 times negative 1 thats why I say its important to wrap the parentheses to make sure you carry the signs through because here now you can see that you have on the top you have to but on the bottom you have negative one times negative one is positive one but times another negative one gives you negative one so the answer you give is actually negative two for the answer and this function evaluated a negative one is negative two and thats the final answer so in this lesson we wanted to reinforce eyes the idea of the function being a machine that takes a mathematical machine that takes input values performs the calculation whatever the function happens to be thats the calculation thats done and then you get outputs that we call f of X so here we just started with very simple functions working our way through the complexity mostly getting practice with substituting in and doing the algebra but this is how functions are evaluated now we have another lesson here were going to have slightly more complex functions but in the next lesson well have exactly the same concept of plugging this guy in and evaluating so make sure you can do all of these problems and then follow me on to the next lesson where we will continue evaluating functions
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