– WE WANT TO FIND THE
COMPOSITION FUNCTIONS F OF G AND G OF F, GIVEN F OF X=X SQUARED
– 2X + 3 AND G OF X=2X + 1. BY DEFINITION,
F OF G=F OF G OF X. SO IN THIS CASE,
NOTICE HOW THE FUNCTION G BECOMES THE INPUT
INTO FUNCTION F, AND G OF F=G OF F OF X, WHICH MEANS F OF X WILL BE
THE INPUT INTO FUNCTION G. SO THE ORDER OF THE COMPOSITION
DOES MATTER. SO F OF G=F OF G OF X, AND JUST LIKE THE ORDER
PARENTHESES, AND SINCE THERE’S NO VALUE
TO SUB INTO FUNCTION G, WE’RE GOING TO SUBSTITUTE 2X + 1
FOR G OF X. SO WE’LL HAVE F OF THE QUANTITY
2X + 1, WHICH MEANS WE’RE GOING
TO SUBSTITUTE 2X + 1 INTO F WHEREVER WE SEE AN X. AND SINCE F OF X
=X SQUARED – 2X + 3, WE’LL HAVE THE QUANTITY
2X + 1 SQUARED – 2 x THE QUANTITY 2X + 1 + 3. AGAIN, NOTICE EVERY X AND F HAS BEEN REPLACED
WITH THE QUANTITY 2X + 1. NOW WE’RE GOING TO MULTIPLY
ALL OF THIS OUT AND COMBINE LIKE TERMS. SO WE’LL HAVE THE QUANTITY
2X + 1 x THE QUANTITY 2X + 1 – 2 x QUANTITY 2X + 1 + 3. SO HERE WE’LL HAVE FOUR
PRODUCTS: ONE, TWO, THREE, FOUR, AND HERE WE’LL THINK OF
DISTRIBUTING A -2 HERE AND HERE. SO WE’LL HAVE 2X x 2X. THAT’S 4X SQUARED
AND THEN 2X x 1–THAT’S 2X, AND THEN 1 x 2X IS ALSO 2X. WE HAVE + 4X,
AND THEN 1 x 1=1 – 4X – 2 + 3, AND NOW WE’LL COMBINE
LIKE TERMS. WE ONLY HAVE ONE X SQUARED TERM. THAT’S 4X SQUARED, AND THEN —
WE HAVE + 4X – 4X, THAT’S 0. SO THERE’RE NO X TERMS,
AND THEN WE HAVE 1 – 2. THAT’S -1 + 3 WOULD BE +2. SO WE HAVE + 2. SO F OF G OF X OR F OF G
=4X SQUARED + 2. NOW LET’S DETERMINE G OF F,
WHICH IS EQUAL TO G OF F OF X, AND AGAIN, STARTING
WITH THE INNERMOST PARENTHESES WE DON’T HAVE A VALUE
TO SUB INTO F. SO WE’RE GOING TO REPLACE F OF X
WITH X SQUARED – 2X + 3. SO THIS’LL BECOME G OF THE
QUANTITY X SQUARED – 2X + 3. SO ALL OF THIS WILL BE THE INPUT
INTO FUNCTION G WHERE G OF X=2X + 1. SO WE’LL HAVE 2 x THE QUANTITY
X SQUARED – 2X + 3, AND DON’T FORGET
WE STILL HAVE THIS + 1. AGAIN, THE X AND G HAS BEEN
REPLACED WITH THE QUANTITY X SQUARED – 2X + 3,
WHICH IS F OF X, AND NOW, WE’LL DISTRIBUTE
AND COMBINE LIKE TERMS. SO WE’LL HAVE
2X SQUARED – 4X + 6 + 1. SO WE HAVE 2X SQUARED – 4X + 7, AND AGAIN, THIS WOULD BE
G OF F OF X OR JUST G OF F. SO WE CAN EASILY SEE THE ORDER
OF THE COMPOSITION DOES MATTER, BECAUSE THE RESULT
OF THESE TWO COMPOSITIONS ARE DIFFERENT FUNCTIONS. REMEMBER, THESE COMPOSITIONS WOULD ONLY BE EQUAL
TO EACH OTHER AND EQUAL TO X IF F AND G
WERE INVERSE FUNCTIONS.