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- WE WANT TO DETERMINE THE COMPOSITE FUNCTION VALUES
USING THE PROVIDED TABLE.
NOTICE OUR COMPOSITE FUNCTIONS ARE WRITTEN USING THIS NOTATION
HERE ON THE LEFT, WHICH IF WE WANT TO,
WE CAN REWRITE AS THE NOTATION GIVEN HERE ON THE RIGHT.
LET'S GO AHEAD AND DO THAT TO BEGIN WITH.
SO WE CAN WRITE F OF G OF -2 USING THIS NOTATION HERE.
G OF F OF 3 CAN BE WRITTEN USING THIS NOTATION HERE.
F OF F OF 0 CAN BE WRITTEN LIKE THIS.
AND G OF F OF -4 CAN BE WRITTEN LIKE THIS.
WHEN EVALUATING A COMPOSITE FUNCTION
IT'S IMPORTANT TO RECOGNIZE
WHICH FUNCTION TO EVALUATE FIRST.
WHEN IT'S WRITTEN USING THIS NOTATION HERE,
IT'S PROBABLY EASIER TO RECOGNIZE
THAT WE'LL ALWAYS START WITH THE INNER FUNCTION,
OR IN THIS CASE, G OF -2.
ANOTHER WAY TO REMEMBER THIS IS TO READ THIS AS F AFTER G OF -2,
WHETHER IT'S IN THIS FORM HERE OR THIS FORM HERE.
SO TO EVALUATE THIS COMPOSITE FUNCTION,
WE'LL FIRST DETERMINE G OF -2.
SO WE GO OVER TO OUR TABLE, FIND AN INPUT OF X VALUE OF -2,
WHICH IS HERE, AND THEN FIND THE OUTPUT FOR FUNCTION G,
WHICH IS 1.
SO WE CAN REPLACE G OF -2 WITH 1.
SO THIS = F OF 1.
NOW WE'LL GO BACK TO OUR TABLE AND FIND THE X VALUE OF 1,
AND FIND THE OUTPUT FOR FUNCTION F.
SO HERE'S X = 1, AND FOR FUNCTION F THE OUTPUT IS 3.
SO THIS = 3, WHICH MEANS F OF G OF -2
ARE OUR COMPOSITE FUNCTION = 3
WHEN THE INITIAL INPUT IS X = -2.
NOW WE HAVE G OF F OF 3,
SO WE'LL FIRST DETERMINE THE VALUE OF F OF 3.
SO WE'LL FIND AN X VALUE OF 3, WHICH IS HERE.
AND THE OUTPUT FOR FUNCTION F IS 4,
SO WE'LL REPLACE F OF 3 WITH 4 GIVING US G OF 4.
TO EVALUATE G OF 4, WE'LL FIND THE INPUT OF X VALUE OF 4,
WHICH IS HERE.
AND FOR FUNCTION G THE OUTPUT IS 2,
WHICH IS THE VALUE OF A COMPOSITE FUNCTION WHEN X = 3.
SO WE CAN WRITE THIS AS G OF F OF 3 = 2.
NEXT WE HAVE F OF F OF 0,
SO WE'LL FIRST DETERMINE THE VALUE OF F OF 0.
SO WE'LL FIND THE OUTPUT FOR FUNCTION F WHEN X = 0.
WELL, HERE'S X = 0, HERE'S FUNCTION F,
SO THE OUTPUT IS 1.
SO WE'LL SUBSTITUTE 1 FOR F OF 0 GIVING US F OF 1,
WHICH WE ALREADY FOUND EARLIER.
WHEN X = 1 THE OUTPUT FOR FUNCTION F IS 3.
SO, AGAIN, WE CAN SAY F OF F OF 0 = 3.
AND THEN FOR OUR LAST EXAMPLE G OF G OF -4,
WE'LL FIRST DETERMINE G OF -4.
WHEN X IS -4 THE OUTPUT FOR FUNCTION G IS 3.
SO THIS WOULD BE = G OF 3.
AND THEN WHEN X IS 3 THE OUTPUT OF FUNCTION G IS 0.
SO WE CAN SAY G OF G OF -4 = 0.
I HOPE YOU FOUND THIS HELPFUL.