Evaluating composite functions: using tables

so we have some tables here that give us what the function f with the functions F and G are when you give it certain inputs so when you input negative for F of negative 4 is 29 that's going to be the output of that function and so we have that for both F and G and what I want to do is evaluate two composite functions I want to evaluate F of G of zero and I want to evaluate G of F of zero so like always pause the video and see if you can figure it out well let's first think about F of G of 0 F of G of 0 what is this all about and actually let me let me use multiple colors here F of G of 0 well this means that we're going to evaluate G at 0 so we're going to input 0 into G let me do it in that so we're going to input 0 into our function G and we're going to output whatever we output is going to be G of 0 I'll write it right over here and then we're going to input that into our function f we're going to input that into our function f and whatever I output then is going to be F of G of 0 F of G of 0 F of G of 0 I wrote these small here so we have space for the actual values so first let's just evaluate and if this if you are now inspired pause the video again and see if you can if you can solve it although if you solved it the first time you don't have to do that now well what's G of 0 well when we input x equals 0 we get G of 0 is equal to 5 so G of 0 is 5 so that is 5 so we're now going to input 5 into our function f we're essentially going to evaluate F of 5 so when you input 5 into our function do it in this brown color when you input x equals 5 into our f you get the function f of or you get F of 5 is equal to 11 so this is going to be 11 so f of G of 0 is is equal to 11 now let's do G of f of zero so now let's evaluate I'll do this in different colors G maybe I'll use those same two colors actually so now we're going to evaluate G of f of zero G of F of zero and the the key realization is you want to go within the parentheses to evaluate that first so then you can evaluate the function that's kind of on the on the outside so here we're going to take zero as an input into the function f and then whatever that is that f of zero we're going to input into our function G we're going to input into our function G and what we're going to be and then the output of that is going to be G of f of zero so let's see what is f of zero well you see over here when you take when our input is zero this table tells us that F of zero is equal to one so f of 0 is equal to 1 f of zero is equal to 1 so now we use 1 as an input into G we're now evaluating G of 1 so I can just write this this is the same thing as G of 1 G of 1 once again why was that because F of 0 is equal to F of 0 is equal to 1 and let me have those parentheses too far away from the G this is the same thing as G of 1 because once again F of 0 is 1 now what is G of 1 well when I input when I will input 1 into our function G I get G of 1 is equal to 8 so this is going to be equal this is equal to 8 and we're done and notice these are different values because these are different composite functions F of G of 0 is 11 and G of F of 0 is 8