Functional Notation

in this video we're gonna talk about functional notation now this is something that I am pretty sure you've seen before but let's look at an example we have f of x equals x squared plus 1 so the way we say it is f of x okay and this does not mean f times x but f of x is equal to x squared plus one not here f is the name of the function x is the input and x squared plus one is the output so for example if we input the number 5 into the function what do we get? well what we do is we replace all of the x's in this formula there's only one x in the formula here but we replace that x with 5 and notice I've done that down here f of 5 is equal to 5 squared plus one or 25 plus one which is 26 so we say f of five is equal to 26 okay 5 is the input and 26 is the output let's look at another example here we have g of x so g is the name of the function is equal to - x^2 + 7x - 2 now let's find g of 5 so again what we need to do is plug in 5 everywhere we see an x we have minus the quantity 5 squared, plus 7 times 5, minus 2 so remember we do the five squared first then we do minus this is not the same thing as (minus x) squared it's minus (x^2) so what we have is minus 25 plus 35 minus 2 now -25 plus 35 is really 10 so 10-2 turns out that it's equal to 8 okay g of 5 is equal to 8 now I'm using minus three here because I wanna do an example where we plug in a negative number so what we have is negative -3 squared plus seven times -3 minus 2 and that would be equal to negative 9 because -3 squared as positive 9 but we have this negative out in front minus 21 and then minus 2 now -9 minus 21 would be -30 and then we subtract another 2 we get -32 okay not here in part (c) I I'm not plugging in a number I'm plugging in a variable but again it's the same procedure we replace all of the x in the formula with a in this case okay so g of a would be negative a squared plus 7a minus 2 alright and the last one for this example is g of x plus h okay what is g of x+h? so again we all of the x's in this formula and we replace them with x+h so what we have is g(x) is equal to -(x+h)^2 plus 7 times (x+h) minus 2 okay I have replaced each of these x's with (x+h) okay now a common mistake that people will make is sometimes they'll just put in the formula for g(x) and then tack-on +h on the end but that's not what you do okay well let's simplify this this really is negative (x+h) times (x+h) plus 7x+7h -2 okay I distributed the 7 now let's FOIL out what (x+h) times (x+h) is that's x squared (x times x is x^2) and we have an xh if we multiply the two outside ones and another xh if we multiply the two inside ones so that's 2xh and finally when we multiply the two last ones here h times h is h^2 okay so our finally answer is minus x^2 minus 2xh minus h^2 I distributed the negative symbol plus 7x plus 7h minus 2 okay so that is g(x+h) okay now one last thing I wanna give a little caution about is this for a function f is it always true that f(a+b) is equal to f(a)+f(b)? okay is that true? it turns out the answer is No ok and again we're not multiply it's not f times (a+b) if this meant f times (a+b) then yes this would be true but this is definitely not always the case in fact most times it's not the case that f(a+b) is equal to f(a)+f(b) so as an example let's look at f(x) equals x^2 okay if you plug in 2 into this formula you get 4 as an output so f(2) is 4 also f(3) is 9 now what is f(2+3)? well f(2+3) is the same thing as f(5) okay 2+3 is 5 and f(5) is 25 so is it the case that f(2+3) is equal to f(2)+f(3)? well f(2+3) that was 25 but this f(2) is 4 f(3) is 9 is it the case that 25 as equal to 4+9? no okay they're not equal to each other so f(a+b) is usually not equal to f(a)+f(b)