– 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

OF OPERATIONS, WE’LL START WITH THE INNERMOST

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.

Not easy to follow along with I got lost and I am in college algebra