Where Students Make Mistakes Evaluating Functions

so on a problem like this students usually do not have trouble being able to figure out what fa2 is they usually have trouble when we're trying to figure out what is f of a Negative X or what is an f of x plus one so in this video that's what I want to focus on again before we can get to these two we got to make sure we understand what exactly does f of 2 equal when we have this function x squared minus 3x plus 4. and to be able to understand this let's kind of go back to like good old days let's just say I have an expression you know 3x plus 1. and let's say let x equal a 5. so this would be a common problem and that we maybe would start off like with an algebra 1 or in pre-algebra and would say you know 3x plus 1 and let x equal five and so what that means is you can just replace the 5 with the X right we're allowing this unknown X to have a value of 5. therefore when I plug a 5 in for the X it just looks like this and so now always to simplify this operation or simplify this expression is just make sure you apply or follow the order of operations right we're going to multiply before we add so in this case I'll have a 3 times 5 which is 15 plus 1 is equal to 16. so that would be my final answer for a problem like that that's really all we're doing when we have a problem like this we have an expression which is going to be what we call our function right this is going to be the equation of our function this F remember is just the name of the function and X is going to represent our input value what we can plug in to our equation so in this case we have f of x right that's the input variable and then our expression x squared minus 3x plus 4. so when we say like f of 2 what we're really saying is like the same thing we said over here let x equal 2 right let the input variable represent as 2. all I need to do if I want to find the value of my function f for the value of my input value 2 I just need to replace X which is the input variable for a value of 2. and now again as I reminded you just make sure that you follow your order of operations we're going to square the 2 squared this will be 4. we can actually multiply this it's not going to impact it so minus 6 and then plus four and then but remember whenever you are simplifying expression with addition and subtraction or double subtraction or double addition just make sure you go from left to right so 4 minus 6 is going to be negative 2 negative 2 plus 4 going to equal a positive 2. so again what this reads is the value of my function f at 2 is guess what is equal to two a lot of times students like a problem like this because what's usually happening is we have a value of a function just like we have the value of the expression over here I have an input value and then I have an output value now I didn't really mean for the output and the input value to be the exact same but it's okay they can now what happens here though is now my input value is another variable we're not going to get an actual value like we did over here but again the operation or the idea is exactly the same this is my new input this is what I'm going to plug in for X notice what I did over here I replace an X with a 2 I put parentheses around it what that did is that reminded me that I am replacing an X with a new input value even though this is still an X I'm still going to do the exact same thing when I plug in Negative X in for X I'm going to put parenthesis around it so I can remember to apply the operations correctly so here's what I want you to see and understand about this because if you don't use parentheses this is where you're going to get the problem wrong because what's usually going to happen is students will not do parentheses around here and they'll do negative x squared equals a negative x squared that's not correct negative x squared represents Negative X times negative X that's actually going to give us a positive X and then over here a lot of times sometimes students will forget this and they'll forget to put the negative and they'll just leave this as a negative 3x this is negative 3 times a negative X that's also going to be a positive X plus four so now my expression here in this case is not a value it's not a number like it was up here because I couldn't replace a number for all these X's I replaced a Negative X but now my expression is going to be an x squared plus 3x Plus 4. so what about when we have something else now we're having X plus a 1. so it's kind of like a combination here of a number and a variable well ladies and gentlemen the exact same thing just plug it in you're going to replace x with an X plus one now here though we need to be even more careful with our parentheses because right negative x squared represents Negative X times negative X well guess what X Plus 1 squared is going to be X Plus 1 times X Plus 1. so now what we need to do is expand this out and remember you can use foil or just you know rewrite it multiplying if you can I'm going to go and do my math a little bit in my head just so you guys can see and I'm going to apply distributive property so all I did was the X Plus 1 squared again it's not x squared plus 1 squared don't do that actually I multiplied it out I did a let me actually just write this out X Plus 1 Quine squared okay so that's what I do over there just wanted to make sure we're all on the same page um now what I need to do is just combine my like terms so I can combine these x's and then I can combine these numbers so therefore I'm going to have a final answer here of an x squared minus X let's do so there go ladies and gentlemen hopefully this video gave you some value in being able to evaluate functions for either a number or a variable expression and if it did give you some value then I know you're going to find some value in my next video