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X
- WE'RE GIVEN F OF X AND G OF X
AND ASKED TO DETERMINE
THE FOLLOWING COMPOSITION OF FUNCTIONS.
THE IDEA OF A COMPOSITION OF FUNCTIONS
IS THAT THE OUTPUT OF THE INNER FUNCTION
BECOMES THE INPUT OF THE OUTER FUNCTION
GIVING US A FINAL OUTPUT FOR THE COMPOSITION OF FUNCTIONS.
TO HELP VISUALIZE THIS WE CAN THINK OF THIS AS A CONVEYER BELT
WHERE IF WE START WITH AN INPUT FOR THE INNER FUNCTION
THIS GIVES US AN OUTPUT.
THIS OUTPUT BECOMES THE INPUT FOR THE OUTER FUNCTION,
WHICH GIVES US A FINAL OUTPUT
FOR THE COMPOSITION OF FUNCTIONS.
SO LOOKING AT OUR FIRST EXAMPLE,
THERE ARE TWO MAIN WAYS
TO EXPRESS A COMPOSITION OF FUNCTIONS AS WE SEE HERE.
THESE TWO ARE EQUIVALENT AND WE SAY F OF G OF 7 IN BOTH CASES.
BUT WHEN EVALUATING A COMPOSITION OF FUNCTIONS
THIS NOTATION HERE IS USUALLY MORE HELPFUL.
WE'RE GOING TO START WITH THE INNER MOST FUNCTION
AND EVALUATE G OF 7.
SO TO EVALUATE G OF 7 WE'LL SUBSTITUTE 7 FOR X IN FUNCTION G
WHICH IS HERE.
SO WE'D HAVE (6 x 7 SQUARED) - (4 x 7) + 5.
WELL 7 SQUARED IS 49, 49 x 6 = 294.
SO WE HAVE (294 - 28) + 5 WHICH IS EQUAL TO 271.
AND NOW, SINCE G OF 7 = 271,
F OF G OF 7 IS EQUAL TO F OF 271.
SO NOTICE HOW THE OUTPUT OF G OF 7
BECOMES THE INPUT INTO FUNCTION F.
SO THAT WOULD GIVE US (2 x 271) - 3
WHICH WOULD BE 542 - 3 OR 539.
SO F OF G OF 7 = 539.
NEXT, WE HAVE G OF F OF 2.
WE'LL BEGIN BY DETERMINING F OF 2 THE INNER FUNCTION VALUE.
SO F OF 2 WOULD BE = (2 x 2) - 3.
WELL THAT'S JUST 4 - 3 OR 1
AND AGAIN SINCE F OF 2 IS EQUAL TO 1,
WE CAN REWRITE G OF F OF 2 AS G OF 1.
SO NOW WE'LL SUBSTITUTE 1 INTO FUNCTION G
TO FIND THE FINAL OUTPUT OF THIS COMPOSITION OF FUNCTIONS.
WE'D HAVE (6 x 1 SQUARED) - (4 x 1) + 5,
WELL THAT WOULD BE 6 - 4 THAT'S 2.
2 + 5 IS 7.
SO G OF F OF 2 = 7.
NOTICE HOW IT IS IMPORTANT THAT WE PAY CLOSE ATTENTION
TO THE ORDER OF THE COMPOSITION.
NEXT, WE HAVE F OF G OF -3,
SO BEGIN BY DETERMINING THE VALUE OF THE INNER FUNCTION,
G OF -3,
WHICH WOULD BE (6 x -3 SQUARED) - (4 x -3) + 5.
WELL -3 SQUARED IS 9, 9 x 6 = 54,
AND THIS WOULD BE + 12 + 5.
SO THIS IS 71.
SINCE G OF -3 = 71
WE CAN REWRITE F OF G OF -3 AS F OF 71.
SO 71 BECOMES THE INPUT INTO FUNCTION F WHICH IS (2 x 71) - 3
WHICH IS 142 - 3 OR 139.
SO F OF G OF -3 = 139.
AND FOR OUR LAST EXAMPLE NOTICE HOW WE HAVE F OF F OF -4.
WE BEGIN BY DETERMINING THE INNER FUNCTION VALUE F OF -4
WHICH WOULD BE (2 x -4) - 3, THAT'S -8 - 3 OR -11.
AND SINCE THE INNER FUNCTION VALUE F OF -4 = -11,
F OF F OF -4 = F OF -11.
SO -11 IS NOW THE INPUT INTO FUNCTION F
WHICH WOULD BE (2 x -11) - 3
THAT WOULD BE -22 - 3 THAT'S - 25.
SO F OF F OF -4 = -25.
I HOPE YOU FOUND THIS HELPFUL.