Evaluating composite functions (advanced)

we're told that H of X is equal to 3x G of T is equal to negative 2 t minus 2 minus H of T f of n is equal to negative 5 n squared plus h of n so we have 3 function definitions and two of these function definitions are actually defined in terms of another function in particular in terms of the function H and then we're asked to calculate what is H of G of 8 and this can be very daunting but we just have to remember that all a function is is something that takes an input and gives you an output for it and this is another way this this somewhat convoluted looking statement is another way of saying look we're going to take the number 8 we're going to take the number 8 and we are going to input it into the function G we are going to input it into the function G and then that is going to produce G of 8 that is going to produce G of 8 and then we're going to take whatever value that is and input that into the function H so we're going to take this whole thing and they're going to input that into the function H and what we will have will be H of H of what we inputted H of G of H of G of 8 so let's just do it one step at a time let's figure out what G of 8 is so G of 8 g of an I'm going to color code it so we can keep track of things G of 8 is equal to well G of T we have our definition here so our input now instead of eight is going to be our T so our input is 8 so every place where we see a t in this function definition we replace it with an 8 so it's going to be negative 2 times 8 negative 2 times 8 minus 2 minus 2 minus and this might be a little daunting but let's just replace this T with an 8 and then see if we can make sense of it h minus let me do it in the right color minus 2 minus 2 minus H of H of 8 notice to evaluate G of 8 all we did is everywhere we saw T we replaced it with the in eight now let's see if we can calculate this this is going to be equal to negative two times eight is negative 16 minus two is negative 18 do that same so this is going to be equals negative 18 minus what is H of 8 going to be equal to so let's do that over here so H of 8 this thing H of 8 now we go to the definition of H don't worry about later we're going to input all this business into H again just let's worry about it one step at a time we need to calculate H of 8 so H of 8 is just going to be well every time we see an X we replace it with an 8 it's going to be 3 times 8 which is equal to 24 so this value right over here is 24 we are subtracting it so we have minus minus 24 negative 18 minus 24 is what that's negative 42 so this all of this business is going to be equal to did I do that right yeah negative negative 42 so we figured out what G of 8 is it is negative 42 so this right over here is negative 42 and now we can input negative 42 into H so H of H I've let me do it right over here H of negative 42 H of negative 42 remember negative 42 is the same thing as G of 8 so this is H of G of 8 is the same thing as H of negative 42 this is going to be equal to that same color this is going to be equal to 3 3 times negative 42 3 times negative 42 which is equal to which is equal to this is negative 126 negative 126 and we are done so it seemed convoluted at first but if you just keep track of what's our input what's our output and just really just evaluate the functions it should hopefully be reasonably straightforward