– 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
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.