- WELCOME TO AN EXAMPLE OF A COMPOSITION OF THREE FUNCTIONS.
GIVEN FUNCTION F, G, AND H
WE WANT TO DETERMINE F OF G OF H OF X,
WHICH WE CAN WRITE USING THIS NOTATION HERE
OR THIS NOTATION HERE.
I PREFER THE NOTATION ON THE RIGHT
BECAUSE IT'S EASIER TO TELL
WE'RE GOING TO START WITH THE INNER MOST FUNCTION
AND WORK OUR WAY OUT.
BUT REMEMBER WE CAN ALSO THINK OF THIS AS
F AFTER G AFTER H OF X.
IN THIS COMPOSITION WE DON'T HAVE AN INPUT FOR FUNCTION H,
SO TO GO AHEAD AND REPLACE H OF X
WITH THE SQUARE ROOT OF X.
SO THIS IS GOING TO BE EQUAL TO F OF G,
AND WE'RE REPLACING H OF X WITH THE SQUARE ROOT OF X.
NOW THAT WE'VE PERFORMED THE FIRST SUBSTITUTION,
NOTICE HOW THE INNER FUNCTION IS G OF THE SQUARE ROOT OF X.
SO NOW WE'RE GOING TO GO TO FUNCTION G
AND REPLACE X WITH THE SQUARE ROOT OF X.
SO THIS WOULD GIVE US F OF--
REPLACING X WITH THE SQUARE ROOT OF X HERE.
WE WOULD HAVE 2 x THE SQUARE ROOT OF X SQUARED,
AND THIS SIMPLIFIES TO JUST 2X.
SO NOW WE WANT TO DETERMINE F OF 2X,
WHICH MEANS WE'RE GOING TO REPLACE X WITH 2X IN FUNCTION F.
SO THIS WOULD GIVE US 2X TO THE THIRD - 2X
AND THEN + 1.
SO THIS WILL GIVE US 2 TO THE THIRD IS 8,
X TO THE THIRD - 2X + 1.
SO THIS WOULD BE OUR COMPOSITE FUNCTION F OF G OF H OF X.
NOW LET'S TAKE A LOOK AT A SECOND EXAMPLE
WHERE WE'RE GIVEN A VALUE OF X FOR THE INITIAL INPUT.
SO HERE WE HAVE THE SAME COMPOSITE FUNCTION,
BUT WE'RE GIVEN THE INITIAL VALUE OF X IS = TO +4.
SO FOR THIS COMPOSITION
WE'RE GOING TO ACTUALLY FIND A NUMERICAL VALUE
FOR THIS COMPOSITE FUNCTION.
AND THERE'S ACTUALLY TWO WAYS OF DOING THIS.
SINCE WE ALREADY DETERMINED THE COMPOSITE FUNCTION
ON THE PREVIOUS SLIDE, OR THIS FUNCTION HERE,
WE CAN ACTUALLY JUST SUBSTITUTE 4 FOR X
INTO OUR COMPOSITE FUNCTION,
WHICH WOULD GIVE US 8 x 4 TO THE THIRD - 2 x 4 + 1.
WELL, 4 TO THE THIRD IS = TO 64,
64 x 8 IS 512 - 8 + 1 IS = TO 505.
SO F OF G OF H OF 4 IS = TO 505.
NOW, I WOULD LIKE TO SHOW A SECOND METHOD
FOR DETERMINING THIS COMPOSITE FUNCTION VALUE.
FOR EXAMPLE,
LET'S SAY WE DIDN'T HAVE THIS COMPOSITE FUNCTION HERE.
WE PROBABLY WOULDN'T DETERMINE THE COMPOSITE FUNCTION
AND THEN SUBSTITUTE 4 IN FOR X.
WE WOULD PROBABLY TAKE A DIFFERENT APPROACH,
WHICH I'LL SHOW NOW.
WE'LL START WITH F OF G OF H OF 4,
AND WE'LL START BY DETERMINING THE VALUE OF H OF 4.
WELL, IF WE SUBSTITUTE 4 FOR X IN FUNCTION, H
WE WOULD HAVE THE SQUARE ROOT OF 4, WHICH IS = TO 2.
SO WE CAN REPLACE H OF 4 WITH 2,
WHICH WOULD GIVE US F OF G OF 2.
NOW WE'D EVALUATE G OF 2.
WELL, IF WE SUBSTITUTE 2 INTO FUNCTION G,
WE'D HAVE 2 x 2 SQUARED OR 2 x 4, WHICH IS = TO 8.
SO THIS WOULD SIMPLIFY TO F OF 8,
AND THEN WE'D REPLACE X WITH 8 IN FUNCTION F,
WHICH WOULD BE A TO THIRD - 8 + 1.
WELL, 8 TO THE THIRD IS 512 - 8 + 1.
NOTICE HOW THE VALUE IS THE SAME, WHICH IT SHOULD BE.
F OF G OF H OF 4 IS = TO 505.
SO OF COURSE, WHICHEVER METHOD WE USE,
THE VALUE OF THE COMPOSITE FUNCTION WHEN X = 4
WILL BE THE SAME.