- WE WANT TO DETERMINE THE COMPOSITE FUNCTION VALUES
FROM THE GRAPHS OF FUNCTION F IN BLUE AND FUNCTION G IN RED.
FROM THIS DEFINITION HERE,
WE SEE THERE ARE TWO WAYS TO EXPRESS A COMPOSITE FUNCTION.
I THINK IT'S GOING TO BE HELPFUL
TO REWRITE THESE COMPOSITE FUNCTIONS USING A NOTATION
GIVEN HERE ON THE RIGHT.
SO FOR F OF G OF 1 OR F AFTER G OF 1,
WE CAN WRITE THIS USING THIS NOTATION HERE.
FOR G OF F OF 9 OR G AFTER F OF 9,
WE CAN WRITE USING THIS NOTATION HERE.
FOR F OF F OF 0, WE CAN WRITE LIKE THIS.
AND FOR G OF G OF 8, WE CAN WRITE LIKE THIS.
THE IMPORTANT THING TO REMEMBER
WHEN EVALUATING A COMPOSITE FUNCTION IS TO KNOW
WHICH FUNCTION TO EVALUATE FIRST.
USING THIS NOTATION HERE,
I THINK IT'S EASIER TO SEE WE'RE ALWAYS GOING TO EVALUATE
THE INNER FUNCTION FIRST.
SO FOR THIS COMPOSITE FUNCTION
WE'LL DETERMINE THE VALUE OF G OF 1 FIRST.
AND ANOTHER WAY TO REMEMBER THIS WOULD BE TO READ THIS
AS F AFTER G OF 1.
SO TO DETERMINE THE VALUE OF G OF 1,
WE NEED TO FIND THE Y COORDINATE ON G WHEN X = 1.
SO HERE'S THE GRAPH OF G, HERE'S X = 1,
SO WE GO STRAIGHT TO THE FUNCTION.
WHEN X IS 1, NOTICE THE CORRESPONDING Y VALUE
OR FUNCTION VALUE WAS 8.
SINCE G OF 1 = 8, WE CAN WRITE THIS AS F OF 8.
AND NOW WE'LL GO TO THE GRAPH OF FUNCTION F
TO DETERMINE THE Y COORDINATE WHEN X = 8.
SO HERE'S THE GRAPH OF F, HERE'S X = 8.
WHEN X IS 8, NOTICE THE CORRESPONDING Y VALUE IS 5.
SO F OF 8 = 5, SO WE CAN SAY F OF G OF 1 = 5.
NEXT WE HAVE G OF F OF 9,
SO WE'LL START BY EVALUATING F OF 9.
SO WE'LL GO TO THE GRAPH OF FUNCTION F
AND DETERMINE THE Y COORDINATE WHEN X IS 9.
HERE'S THE GRAPH OF F, HERE'S X = 9.
NOTICE THE CORRESPONDING Y VALUE OR FUNCTION VALUE IS 3.
SINCE F OF 9 = 3, THIS SIMPLIFIES TO G OF 3.
AND NOW WE'LL GO TO THE GRAPH OF FUNCTION G,
DETERMINE THE Y VALUE WHEN X IS 3.
HERE'S X = 3, CORRESPONDING Y VALUE IS 6,
WHICH IS OUR FUNCTION VALUE, SO THIS = 6.
SO G OF F OF 9 = 6.
LET'S TAKE A LOOK AT OUR LAST TWO EXAMPLES.
HERE WE HAVE F OF F OF 0,
AND WE'LL START BY DETERMINING THE VALUE OF F OF 0.
SO WE'LL FIND THE Y COORDINATE WHEN X = 0 FOR FUNCTION F.
NOTICE WHEN X = 0 WOULD BE RIGHT HERE ON THE Y INTERCEPT,
THE CORRESPONDING Y VALUE IS 1.
SINCE F OF 0 = 1, THIS SIMPLIFIES TO F OF 1.
SO USING THE SAME GRAPH,
WE'LL FIND THE Y COORDINATE WHEN X = 1.
SO HERE'S X = 1, CORRESPONDING Y VALUE IS 2.
SO THIS = 2, WHICH MEANS F OF F OF 0 = 2.
AND FOR OUR LAST EXAMPLE WE HAVE G OF G OF 8.
USING THE GRAPH OF FUNCTION G,
WE'LL DETERMINE THE Y COORDINATE WHEN X = 8 TO EVALUATE G OF 8.
SO WHEN X = 8 NOTICE THE FUNCTION VALUE
OR CORRESPONDING Y VALUE IS 2.
SO THIS = G OF 2.
AND THEN USING THE GRAPH OF FUNCTION G
AGAIN, WE'LL FIND THE Y COORDINATE WHEN X = 2.
SO WHEN X IS 2 THE Y COORDINATE
OR CORRESPONDING FUNCTION VALUE IS 7.
SO THIS = 7,
THEREFORE, G OF G OF 8 = 7.
THAT'LL DO IT FOR THIS EXAMPLE. I HOPE YOU FOUND THIS HELPFUL.