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X
- GIVEN FUNCTIONS F AND G,
WE WANT TO DETERMINE THE COMPOSITE FUNCTION F OF G OF X,
AND ALSO G OF F OF X.
FOR THE FIRST STEP,
LET'S REWRITE THIS IN THE FORM OF F OF G OF X.
THIS FORM'S GOING TO BE EASIER TO WORK WITH,
AND WE'LL WRITE G OF F OF X IN THIS FORM HERE.
SO WE'LL START WITH THE INNER FUNCTION.
NOTICE THERE'S NO INPUT INTO FUNCTION G,
SO WE'RE GOING TO REPLACE G OF X WITH 3 DIVIDED BY X - 5.
SO WE CAN WRITE THIS AS F OF 3 DIVIDED BY X - 5.
AND NOW THIS BECOMES THE INPUT INTO FUNCTION F.
SO WE'LL SUBSTITUTE THIS QUANTITY FOR THIS X HERE,
AND THIS WILL GIVE US OUR COMPOSITE FUNCTION.
WE'LL HAVE 1 DIVIDED BY THE QUANTITY 3 DIVIDED BY X - 5,
AND THEN WE STILL HAVE + 5.
WELL, THIS ACTUALLY WORKS OUT PRETTY WELL
BECAUSE NOTICE HOW WE HAVE - 5 HERE + 5 THAT WOULD BE 0.
SO THIS SIMPLIFIES NICELY TO 1/3/X.
REMEMBER THIS FRACTION BAR MEANS DIVISION,
SO IF IT'S HELPFUL,
WE CAN WRITE THIS AS A MULTIPLICATION PROBLEM.
THIS WOULD BE 1 X THE RECIPROCAL OF 3/X, WHICH WOULD BE X/3,
WHICH OF COURSE IS JUST X DIVIDED BY 3.
SO OUR COMPOSITE FUNCTION F OF G OF X IS = TO X DIVIDED BY 3.
NOW LETS TAKE A LOOK AT G OF F OF X.
AGAIN, WE'LL START WITH THE INNER FUNCTION F OF X.
AND THERE'S NO INPUT FOR FUNCTION F,
SO WE'RE GOING TO REPLACE F OF X
WITH 1 DIVIDED BY THE QUANTITY X + 5.
SO THIS IS = TO G OF 1 DIVIDED BY THE QUANTITY X + 5.
AND NOW THIS BECOMES THE INPUT INTO FUNCTION G,
WHICH MEANS WE'LL SUBSTITUTE THIS QUANTITY HERE FOR THE X
IN FUNCTION G.
SO THIS IS = TO 3 DIVIDED BY 1/X + 5 AND THEN - 5.
NOW, FOR THE NEXT STEP, THIS IS 3 DIVIDED BY THIS FRACTION,
WHICH IS THE SAME AS 3 x THE RECIPROCAL OF THIS FRACTION.
SO THIS WOULD BE 3 x THE RECIPROCAL OF 1/X + 5
IS JUST X + 5.
AND THEN WE STILL HAVE THIS - 5 HERE, SO NOW WE'LL DISTRIBUTE.
THIS WOULD BE 3X + 15 - 5, WHICH IS = TO 3X + 10.
SO G OF F OF X IS = TO 3X + 10.
I HOPE YOU FOUND THIS HELPFUL.