Ex 1:  Domain of a Composite Function – No Restrictions
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Ex 1: Domain of a Composite Function – No Restrictions


– GIVEN F OF X AND G OF X, WE WANT TO FIND
THE COMPOSITE FUNCTION F OF G, AND THEN DETERMINE THE DOMAIN OF
THE COMPOSITE FUNCTION F OF G. DO NEED TO BE CAREFUL WHEN DETERMINING THE DOMAIN
OF COMPOSITE FUNCTIONS. THE DOMAIN OF THE COMPOSITE
FUNCTION, F OF G OF X, MUST CONTAIN THE RESTRICTIONS
OF THE DOMAIN OF THE INNER FUNCTION G OF X AND THE RESTRICTIONS
ON THE COMPOSITE FUNCTION. SO IT’S IMPORTANT
TO CONSIDER THE DOMAIN OF THE INNER FUNCTION, AS WELL AS THE
COMPOSITE FUNCTION TO DETERMINE THE CORRECT DOMAIN
OF F OF G OF X. SO TO START OFF, LET’S WRITE
OUR COMPOSITE FUNCTION USING THIS DEFINITION HERE. SO F OF G,
SOMETIMES WRITTEN LIKE THIS, OR EVEN LIKE THIS, CAN BE WRITTEN AS F OF G OF X. USING THIS NOTATION,
WE AN EASILY SEE THAT THE INNER FUNCTION
WILL BE G OF X. SO BEFORE WE GO ANY FURTHER, LET’S DETERMINE THE DOMAIN
OF G OF X. WELL, NOTICE THAT G OF X
IS JUST A LINEAR FUNCTION, WHICH MEANS THERE ARE NO
RESTRICTIONS ON THE DOMAIN, AND THEREFORE THE DOMAIN
WOULD BE ALL REAL NUMBERS. IF WE WANTED TO,
USING INTERVAL NOTATION, WE COULD EXPRESS THIS AS THE
INTERVAL FROM NEGATIVE INFINITY TO POSITIVE INFINITY. SO NOTICE THAT THERE ARE
NO RESTRICTIONS ON THE DOMAIN OF G OF X, WHICH MEANS WE CAN GO AHEAD
AND DETERMINE THE COMPOSITE FUNCTION. AND THE DOMAIN
OF THE COMPOSITION WILL BE THE DOMAIN OF F OF G. AND SINCE G OF X IS=TO 3X – 2 WE’LL PERFORM THIS SUBSTITUTION
FOR G OF X, SO THIS WILL BECOME
F OF QUANTITY 3X – 2. SO NOTICE HOW 3X – 2 BECOMES
THE INPUT INTO FUNCTION F. AND SINCE F OF X IS=
TO THE QUANTITY X + 2 SQUARED, THIS BECOMES THE QUANTITY
3X – 2 + 2 SQUARED. HERE WE HAVE A – 2 + 2
THAT’S=TO 0, SO THIS SIMPLIFIES TO
3X SQUARED, WHICH IS=TO 9X SQUARED. SO OUR COMPOSITE FUNCTION,
F OF G IS=TO 9X SQUARED. NOTICE HOW THIS IS
A QUADRATIC FUNCTION, AND A DOMAIN OF A QUADRATIC
FUNCTION IS ALSO REAL NUMBERS, THEREFORE WE CAN CONCLUDE THE
DOMAIN OF OUR COMPOSITE FUNCTION IS ALL REAL NUMBERS. SO THIS IS ONE EXAMPLE WHERE IF WE DID NOT CONSIDER THE
DOMAIN OF THE INNER FUNCTION, OR G OF X, WE WOULD’VE BEEN OKAY BY JUST FINDING THE DOMAIN
OF OUR COMPOSITE FUNCTION. BUT WHAT WE’LL SEE
IN THE NEXT FEW EXAMPLES IS THIS IS NOT ALWAYS THE CASE. SO IT IS IMPORTANT THAT WE ALSO
CONSIDER THE RESTRICTIONS ON THE INNER FUNCTION.

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