- WELCOME TO A VIDEO ON COMPOSITE FUNCTIONS.
A COMPOSITE FUNCTION CAN BE THOUGHT OF
AS A COMBINATION OF TWO FUNCTIONS.
ONE WAY TO ILLUSTRATE THIS WOULD BE TO USE A CONVEYOR BELT
WHERE WE HAVE AN INITIAL INPUT OF, LET'S SAY, 3,
ONE WAY TO ILLUSTRATE THIS WOULD BE TO USE A CONVEYOR BELT
WHERE WE HAVE AN INITIAL INPUT OF, LET'S SAY, 3,
AND IT BECOMES THE INPUT INTO F WHERE F SQUARES ITS INPUT
AND PRODUCES AN OUTPUT OF 9,
AND THIS OUTPUT BECOMES THIS INPUT INTO G.
SO WE MULTIPLY 2 x 9,
AND THE FINAL OUTPUT OF THIS COMPOSITE FUNCTION
WOULD BE 18.
NOW THE ORDER IN WHICH WE EVALUATE A COMPOSITE FUNCTION
IS EXTREMELY IMPORTANT.
SO FOR EXAMPLE IF WE HAVE THE SAME FUNCTIONS,
BUT NOW WE REVERSE THE ORDER,
AND START WITH THE SAME INPUT OF 3,
FUNCTION G WOULD MULTIPLY THAT BY 2, PRODUCING AN OUTPUT OF 6,
WHICH BECOMES THE INPUT INTO F, AND 6 SQUARED WOULD BE 36.
SO NOTICE IF WE CHANGE THE ORDER IN WHICH WE COMPOSE FUNCTIONS,
THE OUTPUTS IN MOST CASES WILL BE QUITE DIFFERENT
EVEN THOUGH OUR INPUTS MAY BE THE SAME.
THE FORMAL DEFINITION OF A COMPOSITE FUNCTION STATES,
"GIVEN TWO FUNCTIONS F AND G,
"THE COMPOSITE FUNCTION DENOTED USING THIS NOTATION
IS READ F COMPOSED WITH G, OR JUST F OF G."
AND WE NEED TO BE CAREFUL NOT TO CONFUSE THIS SYMBOL HERE
WITH THE MULTIPLICATION SIGN.
NOTICE FOR A COMPOSITION OF FUNCTIONS
IT'S A HOLLOW CIRCLE,
WHEREAS FOR MULTIPLICATION IT WOULD BE SOLID.
BELOW HERE WE SEE THREE EQUIVALENT WAYS
TO REPRESENT A COMPOSITION OF FUNCTIONS.
REMEMBER FOR A COMPOSITE FUNCTION THE ORDER
IN WHICH WE COMPOSE THE FUNCTIONS
IS EXTREMELY IMPORTANT.
THEREFORE, WHEN WE'RE GIVEN A COMPOSITE FUNCTION,
WE'RE GOING TO REWRITE IT IN THIS FORM HERE,
BECAUSE THIS TELLS US THAT WE'RE GOING TO FIRST EVALUATE
FUNCTION G AT THE VALUE OF X,
AND THE OUTPUT OF THAT WILL BE THE INPUT INTO FUNCTION G.
SO WE'RE GOING TO EVALUATE FUNCTION G FIRST,
AND THEN FUNCTION F,
IF THE COMPOSITION OF FUNCTIONS IS WRITTEN IN THIS WAY.
AND THAT'S ALMOST IN THE OPPOSITE DIRECTION
THAT YOU MIGHT THINK.
NOTICE THE F COMES BEFORE THE G, BUT WHEN IT COMES TO EVALUATING,
THE GIVEN VALUE OF X WILL FIRST BE THE INPUT INTO G,
AND THAT OUTPUT WILL BECOME THE INPUT INTO F.
LET'S GO AND LOOK AT SOME EXAMPLES.
IF WE WANT TO EVALUATE F OF G OF 3,
THIS IS GOING TO BE EQUAL TO F OF G OF 3 WRITTEN THIS WAY.
AND SO THIS TELLS US THAT WE WILL FIND
THE VALUE OF G OF 3 FIRST,
AND THAT RESULT WILL BE THE INPUT INTO F.
SO LET'S GO DOWN HERE ON THE SIDE AND FIND G OF 3.
G OF 3 = 3 CUBED - 5, AND THAT WILL BE 27 - 5 = 22.
SINCE G OF 3 = 22, THIS JUST BECOMES F OF 22.
WE FOUND G OF 3, AND IT WAS EQUAL TO 22.
NOW 22 WILL BECOME THE INPUT INTO F.
NOTICE HERE F MULTIPLIES THE INPUT BY 2,
AND THEN SUBTRACTS 1.
SO WE HAVE 44 - 1, SO WE HAVE 43.
SO F OF G OF 3 = 43.
LET'S GO AND TAKE A LOOK AT ANOTHER ONE.
NOTICE THIS QUESTION HAS THE SAME INPUT,
BUT NOW INSTEAD OF F OF G IT'S G OF F.
SO THIS IS EQUAL TO G OF F OF 3,
SO NOW WE'RE GOING TO FIND F OF 3 FIRST.
LET'S GO DOWN HERE AND DO THAT.
F OF 3 = 2 x X - 1 OR 2 x 3 - 1.
THAT WILL BE 6 - 1 OR 5, SO F OF 3 = 5.
SO WE CAN REPLACE THIS WITH 5, SO THIS BECOMES G OF 5.
AND G OF X = X CUBED - 5, SO WE'LL HAVE 5 CUBED - 5.
SO THAT'S 125 - 5 = 120.
SO THIS WOULD BE THE VALUE OF G OF F OF 3.
LET'S TRY ANOTHER.
WE WANT H OF G OF -1.
WELL, THIS IS EQUAL TO H OF G OF -1 WRITTEN THIS WAY,
AND THIS SHOULD KEY US IN TO EVALUATE G OF -1 FIRST.
SO WE'LL GO AHEAD AND DO THIS ON THE SIDE, G OF -1.
WELL, G OF X = X CUBED - 5,
SO WE HAVE -1 CUBED - 5, SO THIS WILL BE -1 - 5 OR -6.
SO G OF -1 = -6, SO NOW WE CAN REWRITE THIS AS H OF -6.
SO NOW WE REPLACE X WITH -6 AND H, SO WE'VE GOT 5 - -6 SQUARED.
WATCHING OUR SIGNS HERE, WE'RE GOING TO HAVE 5 - 36 = -31.
SO H OF G OF -1 = -31.
OKAY, NOW THERE'S ONE OTHER TYPE OF PROBLEM WE NEED TO LOOK AT.
NOTICE ON THIS PROBLEM THEY DON'T GIVE US AN INITIAL INPUT.
SO WHAT WE'RE GOING TO HAVE TO DO IS INPUT ONE OF THE FUNCTIONS
INTO THE OTHER FUNCTION.
AND AGAIN, IT'S IMPORTANT THAT WE FOLLOW THE CORRECT ORDER.
SO REMEMBER THIS IS THE SAME AS F OF G OF X,
WHICH BY DEFINITION IS EQUAL TO F OF G OF X WRITTEN THIS WAY.
NOTICE THERE'S NO X VALUE TO INPUT INTO G,
SO THE ONLY THING WE CAN REALLY DO IS REPLACE G OF X
WITH X SQUARED - X + 5.
SO WE NEED TO DETERMINE F OF X SQUARED - X + 5.
SO IF THERE'S VALUE TO INPUT INTO G,
WE HAVE TO REPLACE G OF X WITH THE ENTIRE FUNCTION.
SO NOW WHEREVER WE SEE AN X IN F
WE'LL REPLACE IT WITH THIS ENTIRE EXPRESSION.
SO INSTEAD OF 4X + 1
WE'LL HAVE 4 x THE QUANTITY X SQUARED - X + 5 + 1.
SO WE'LL DISTRIBUTE THE 4 AND THEN COMBINE IT IN LIKE TERMS.
SO 4X SQUARED - 4X + 20 + 1, SO F OF G = 4X SQUARED - 4X,
AND THEN WE'D HAVE + 21.
SOMETIMES THESE ARE A LITTLE TRICKY WHEN YOU FIRST DO THESE,
BUT THE PROCESS IS EXACTLY THE SAME
EXCEPT INSTEAD OF HAVING A NUMERICAL VALUE
YOU HAVE TO SUBSTITUTE AN ENTIRE FUNCTION
INTO THE OTHER FUNCTION.
LET'S TRY ONE MORE OF THESE.
NOW WE HAVE G OF F INSTEAD OF F OF G,
SO THIS IS EQUAL TO G OF F OF X.
AGAIN, WE DON'T HAVE AN INPUT TO PUT INTO FUNCTION F,
SO WE'RE GOING TO REWRITE THIS AS G OF--WELL, F OF X = 4X + 1.
SO WHEREVER WE SEE AN X IN G WE'LL REPLACE IT WITH 4X + 1.
SO WE'RE GOING TO HAVE, INSTEAD OF X SQUARED,
WE'LL HAVE 4X + 1 SQUARED - 4X + 1 + 5.
HERE WE'RE GOING TO HAVE TO SQUARE 4X + 1.
THERE'S NO SHORTCUTS HERE.
WE'RE GOING TO HAVE 4X + 1 x 4X + 1.
THIS WILL BE - 4X - 1 + 5.
OKAY, SO WE'RE GOING TO HAVE 16X SQUARED + 4X + 4X.
THAT'S + 8X + 1 - 4X.
THIS WILL BE + 4.
AND WE HAVE ONE LAST STEP.
YOU CAN SEE THIS TYPE OF PROBLEM IS A LOT MORE WORK.
WE'RE GOING TO HAVE 16X SQUARED + 4X.
HERE WE HAVE A 1 AND A 4 + 5.
AND NOTICE THAT WHEN WE CHANGE THE ORDER OF THE COMPOSITION
THE RESULT WAS QUITE DIFFERENT
FROM WHAT WE HAD ON THE PREVIOUS SCREEN.
ONE LAST COMMENT I'D LIKE TO MAKE.
IF WE DID DETERMINE F OF G, AND IT WAS EQUAL TO G OF F,
WHICH WAS EQUAL TO X,
THEN THE TWO FUNCTIONS HAVE A SPECIAL RELATIONSHIP,
AND THEY ARE INVERSES OF ONE ANOTHER.
AND THIS TOPIC WILL BE COVERED IN ANOTHER VIDEO.
SO THANK YOU FOR WATCHING, AND HAVE A GOOD DAY.