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- WE'RE GIVEN F OF X AND G OF X
AND ASKED TO DETERMINE
THE FOLLOWING COMPOSITE FUNCTION VALUES.
THE IDEA OF A COMPOSITE FUNCTION
IS THAT THE OUTPUT OF THE INNER FUNCTION
WILL BECOME THE INPUT OF THE OUTER FUNCTION
GIVING US THE FINAL OUTPUT OF THE COMPOSITION OF FUNCTIONS.
TO VISUALIZE THIS WE CAN THINK OF A CONVEYER BELT
WHERE WE START WITH THE INPUT INTO THE INNER FUNCTION
WHICH GIVES US AN OUTPUT.
THEN THIS OUTPUT BECOMES THE INPUT INTO THE OUTER FUNCTION
GIVING US A FINAL OUTPUT FOR THE COMPOSITION OF FUNCTIONS.
SO LOOKING AT OUR FIRST EXAMPLE,
NOTICE HOW WE CAN WRITE COMPOSITE FUNCTION
TWO MAIN WAYS.
THESE TWO FORMS ARE EQUIVALENT
THOUGH THE SECOND FORM IS EASIER TO WORK WITH
BECAUSE WE CAN SEE THE INNER FUNCTION WOULD BE G OF 2.
SO TO FIND THE VALUE OF F OF G OF 2,
WE'LL FIRST DETERMINE G OF 2.
TO DETERMINE THE VALUE OF G OF 2,
WE SUBSTITUTE 2 FOR X IN FUNCTION G,
WHICH WOULD BE 2 x 2 SQUARED, - 5 x 2 + 9.
WELL THIS WOULD BE 4 x 2, THAT'S 8 - 10, THAT'S -2 + 9 THAT'S +7.
SINCE G OF 2 = +7, F OF G OF 2 = F OF 7.
NOTICE HOW THE OUTPUT OF G
BECOMES THE INPUT IN THE FUNCTION F.
F OF X = THREE FIFTHS X + 4
SO F OF 7 WOULD BE THREE FIFTHS x 7 OR x 7/1 + 4 + 4/1.
NOTICE OUR DENOMINATOR HERE IS GOING TO BE 5.
LET'S CONVERT 4/1 TO FIFTHS BY MULTIPLYING BY 5/5.
NOTICE HOW WE HAVE A COMMON DENOMINATOR OF 5 NOW
AND THE NUMERATOR WOULD BE 21 + 20 WHICH WOULD BE 41.
SO F OF G OF 2 = 41 FIFTHS.
NEXT WE HAVE G OF F OF 4.
SO WE'LL BEGIN BY DETERMINING F OF 4
WHICH IS THE INNER FUNCTION VALUE.
SO F OF 4 WOULD BE EQUAL TO THREE FIFTHS x 4 + 4
OR THREE FIFTHS x 4/1 + 4 OR 4/1.
AGAIN NOTICE HOW THE COMMON DENOMINATOR HERE WOULD BE 5.
ONCE AGAIN LET'S CONVERT 4/1 TO FIFTHS.
NOTICE NOW WE DO HAVE A COMMON DENOMINATOR OF 5
AND THE NUMERATOR WOULD BE 12 + 20 OR 32 FIFTHS.
AND SINCE F OF 4 = 32 FIFTHS,
WE CAN RE-WRITE G OF F OF 4 AS G OF 32 FIFTHS.
SO THE INPUT IN A G IS NOW 32 FIFTHS
SO THE OUTPUT WOULD BE 2 OR 2/1 x 32 FIFTHS SQUARED - 5 OR - 5/1
x 32 FIFTHS.
AND THEN + 9 OR + 9/1.
SO LET'S SEE WHAT THIS IS GOING TO WORK OUT TO.
32 SQUARED IS 1,024, 1,024 x 2 WOULD BE 2,048
SO WE HAVE 2,048/5 SQUARED THAT'S 25,
MINUS 160 FIFTHS + 9/1 --
OUR COMMON DENOMINATOR IS GOING TO BE 25.
LET'S MULTIPLY THIS FRACTION BY 5/5,
MULTIPLY THIS FRACTION BY 25/25.
NOTICE HOW NOW WE HAVE A COMMON DENOMINATOR
THE COMMON DENOMINATOR IS 25.
SO WE HAVE 2,048 - 160 x 5 IS 800, SO MINUS 800
AND THEN + 9 x 25 IS EQUAL TO 225.
SO 2,048 - 800 + 225 IS EQUAL TO 1,473.
THIS WOULD BE ALL OVER 25.
THIS WOULD BE G OF F OF 4.
LET'S TAKE A LOOK AT TWO MORE EXAMPLES.
HERE WE HAVE F OF G OF ONE HALF.
WE'LL BEGIN BY DETERMINING G OF ONE HALF.
G OF ONE HALF WOULD BE EQUAL TO 2 OR 2/1 x ONE HALF SQUARED.
MINUS 5 OR 5/1 x ONE HALF
AND THEN + 9 OR + 9/1.
SO HERE WE'D HAVE 2 x ONE FOURTH,
THAT WOULD BE ONE HALF - 5 HALVES.
LET'S GO AHEAD AND CONVERT 9/1 TO HALVES BY MULTIPLYING BY 2/2
SO THAT WOULD BE + 18 HALVES.
DENOMINATOR IS 2, THE NUMERATOR WOULD BE 1 - 5 THAT'S -4.
-4 + 18 WOULD BE 14,
14 DIVIDED BY 2 = 7.
SINCE G OF ONE HALF = 7, F OF G OF ONE HALF IS EQUAL TO F OF 7.
AND F OF 7 WOULD BE EQUAL TO THREE FIFTHS x 7 OR 7/1
+ 4 OR + 4/1.
NOTICE OUR DENOMINATOR HERE IS GOING TO BE 5.
LET'S CONVERT 4/1 TO FIFTHS BY MULTIPLYING BY 5/5.
NOTICE OUR DENOMINATOR IS NOW 5
AND THE NUMERATOR WOULD BE 21 + 20 OR 41 FIFTHS.
SO F OF G OF ONE HALF = 41 FIFTHS.
AND NOW FOR OUR LAST EXAMPLE WE HAVE F OF F OF -FOUR FIFTHS.
WE BEGIN BY DETERMINING F OF -FOUR FIFTHS.
SO F OF -FOUR FIFTHS WOULD BE EQUAL TO THREE FIFTHS
x -FOUR FIFTHS + 4 + 4/1
SO WE'D HAVE -12 TWENTY FIFTHS
AND THEN LET'S CONVERT THIS TO 20 FIFTHS
BY MULTIPLYING BY 25/25.
SO WE HAVE + 100 25TH SO THIS WOULD BE EQUAL TO WHAT 88 25TH.
SO THIS -- F OF NEGATIVE FOUR FIFTHS = 88 25TH
WE CAN RE-WRITE THIS COMPOSITE FUNCTION AS F OF 88 25TH.
SO THIS BECOMES THE INPUT AGAIN INTO FUNCTION F.
SO THIS WOULD BE EQUAL TO THREE FIFTHS x 88/25 + 4 OR 4/1.
SO NOTICE HERE WE'D HAVE A DENOMINATOR OF 5 x 25,
THAT'S 125
3 x 88 = 264.
LET'S MULTIPLY 4/1 BY 125/125
SO THIS WOULD BE + WHAT 500/125
SO WE HAVE THE SUM OF 764, 120 FIFTHS
WHICH WOULD BE F OF F OF -FOUR FIFTHS.
I HOPE YOU FOUND THIS HELPFUL.