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- WE WANT TO DETERMINE THE COMPOSITE FUNCTION VALUES
F OF G OF 0 AND G OF F OF 0,
GIVEN F OF X = 12 + 3X - X SQUARED,
AND G OF X = -3X + 15.
BY DEFINITION, F OF G = F OF G OF X.
SO F OF G OF 0 = F OF G OF 0,
AND THEN TO EVALUATE THIS,
WE'LL START WITH THE INNER MOST FUNCTION,
WHICH MEANS WE'LL FIND G OF 0 FIRST.
SO LET'S GO AHEAD AND DO THAT BELOW,
AND SINCE G OF X = -3X + 15,
G OF 0 IS GOING TO BE EQUAL TO - 3 x 0 + 15.
SO G OF 0 = 15,
AND NOW WE'LL SUBSTITUTE 15 FOR G OF 0.
SO OUR COMPOSITE FUNCTION = F OF 15,
AND NOW WE'LL SUBSTITUTE 15 INTO F
TO DETERMINE THIS COMPOSITE FUNCTION VALUE.
SO WE'D HAVE 12 PLUS 3 x 15 - 15 SQUARED.
SO WE'LL HAVE 12 +-- THIS'LL BE 45
MINUS 15 SQUARED = 225.
12 + 45 = 57 - 225 = -168.
SO F OF G OF 0 = -168.
NOW, FOR THE SECOND EXAMPLE, SINCE G OF F = G OF F OF X,
G OF F OF 0 = G OF F OF 0,
AND AGAIN, STARTING WITH THE INNERMOST FUNCTION,
WE'LL START BY DETERMINING F OF 0.
LET'S GO AHEAD AND DO THAT BELOW.
SINCE F OF X = 12 + 3X - X SQUARED,
WE WOULD HAVE 12 + 3 x 0 - 0 SQUARED.
SO F OF 0 = 12,
AND SINCE F OF 0 = 12, THIS SIMPLIFIES TO G OF 12.
SO NOW, WE'LL SUBSTITUTE 12 INTO G WHERE G = -3X + 15.
SO WE'LL HAVE -3 x 12 + 15.
WELL, THAT'S GOING TO BE -36 + 15, WHICH = -21.
SO G OF F OF 0 = -21.