Ex 3:  Domain of a Composite Function

Ex 3: Domain of a Composite Function


– 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. WE NEED TO BE CAREFUL WHEN DETERMINING THE DOMAIN
OF A COMPOSITE FUNCTION BECAUSE THE DOMAIN OF F OF G OF
X MUST CONTAIN THE RESTRICTIONS OF THE DOMAIN OF THE INNER
FUNCTION G OF X, AS WELL AS THE RESTRICTIONS ON
NEWLY FORM COMPOSITE FUNCTION. SO TO START,
IT’S GOING TO BE HELPFUL TO WRITE OUR COMPOSITE FUNCTION
USING THIS DEFINITION HERE. SO WE CAN WRITE F OF G, SOMETIMES WRITTEN LIKE THIS
OR EVEN LIKE THIS CAN BE WRITTEN IN THE FORM OF
F OF G OF X. WRITTEN IN THIS FORM,
IT’S EASIER TO SEE THAT THE INNER FUNCTION
IS G OF X. SO WHEN CONSIDERING THE DOMAIN
OF OUR COMPOSITE FUNCTION, WE FIRST NEED TO DETERMINE
THE DOMAIN OF G OF X BECAUSE THESE RESTRICTIONS
MUST ALSO BE INCLUDED IN THE RESTRICTIONS
OF OUR COMPOSITE FUNCTION. AND SINCE G OF X IS=TO 1/X, AND WE KNOW DIVISION BY 0
IS UNDEFINED, THE DOMAIN OF GO OF X
ARE INNER FUNCTION WOULD BE ALL REAL NUMBERS EXCEPT X=0. SO THIS MEANS REGARDLESS
OF WHAT OUR COMPOSITE FUNCTION MIGHT LOOK LIKE, WE MUST EXCLUDE 0 FROM THE DOMAIN
IN OUR COMPOSITE FUNCTION. NOW LET’S GO AHEAD AND DETERMINE
OUR COMPOSITE FUNCTION AND SEE IF THERE ARE MORE VALUES
THAT WE MUST EXCLUDE. AGAIN, SINCE G OF X IS=TO 1/X,
WE CAN WRITE THIS AS F OF 1/X. SO NOW THE INPUT
INTO FUNCTION F WILL BE 1/X. SO WHEREVER WE SEE AN X AND F,
WE’LL REPLACE IT WITH 1/X. SO THIS X HERE,
WE REPLACE WITH 1/X. SO WE’LL HAVE 1 OVER– INSTEAD OF X,
WE’LL HAVE 1/X – 4. SO THIS WOULD BE
OUR COMPOSITE FUNCTION, WHICH IF WE WANTED TO ELIMINATE
THIS FRACTION HERE IN THE DENOMINATOR, WE CAN MULTIPLY THE DENOMINATOR
BY X, AS WELL AS THE NUMERATOR TO CHANGE THE FORM
OF THIS FUNCTION. SO LET’S GO AHEAD AND DO THAT. F OF G WOULD BE=TO X ALL OVER
HERE WE’D HAVE 1 – 4X. SO NOW THAT WE HAVE
OUR COMPOSITE FUNCTION, WE SHOULD BE ABLE TO RECOGNIZE THAT WE’D ALSO HAD DIVISION BY 0
IF 1 – 4X WAS=TO 0. SO NOW THAT WE HAVE
OUR COMPOSITE FUNCTION, WE HAVE TO DETERMINE WHAT OTHER
VALUES OF X WE MUST EXCLUDE. SO IN ADDITION TO THE
RESTRICTIONS FROM G OF X, WE ALSO NEED TO KNOW
WHEN 1 – 4X WOULD BE=TO 0. SO IF WE ADD 4X TO BOTH SIDES
OF THE EQUATION WE WOULD HAVE 1=4X. DIVIDE BOTH SIDES BY 4
AND WE HAVE X=1/4. SO THIS VALUE
MUST ALSO BE EXCLUDED FROM THE DOMAIN
OF THE COMPOSITE FUNCTION BECAUSE IT WOULD BE
ANOTHER VALUE WHERE WE’D HAVE DIVISION BY 0. SO THE DOMAIN
OF OUR COMPOSITE FUNCTION WOULD BE ALL REAL NUMBERS EXCEPT X=0 FROM THE DOMAIN OF OUR
INNER FUNCTION G OF X, AS WELL AS X=1/4. THIS WOULD BE THE CORRECT DOMAIN
OF OUR COMPOSITE FUNCTION. IF WE DID WANT TO EXPRESS THIS
USING INTERVAL NOTATION, WE WOULD HAVE TO DRAW
A NUMBER LINE. WE’D EXCLUDE 0 AND EXCLUDE 1/4
WITH OPEN POINTS, AND THEN GRAPH
EVERY OTHER REAL NUMBER. SO WE GRAPH TO THE RIGHT,
IN BETWEEN, AND TO THE LEFT. SO USING INTERVAL NOTATION,
WE’D HAVE THREE INTERVALS FROM NEGATIVE INFINITY TO 0
UNION FROM 0 TO 1/4 UNION, FROM 1/4
TO POSITIVE INFINITY. SO AS YOU KNOW,
THERE’S MORE THAN ONE WAY TO EXPRESS THE DOMAIN
OF A FUNCTION. HERE WE HAVE IT IN WORDS,
HERE WE HAVE THE GRAPH, AS WELL AS INTERVAL NOTATION. OKAY, WE’LL TAKE A LOOK
AT ANOTHER EXAMPLE IN THE NEXT VIDEO.

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