– 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.

Nicely explained!!

where did you get the 4

sorry im a bit slow at math