Ex:  Domain of Composite Function From Graphs
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Ex: Domain of Composite Function From Graphs


– WE WANT TO FIND THE DOMAIN OF THE COMPOSITE FUNCTIONS
F OF G OF X AND G OF F OF X FROM THE GRAPH OF F OF X
AND THE GRAPH OF G OF X. SO BECAUSE WE’RE NOT GIVEN
THE EQUATIONS FOR F OF X AND G OF X, DETERMINING THE DOMAIN
OF OUR TWO COMPOSITE FUNCTIONS CAN BE MORE CHALLENGING. LET’S START BY DETERMINING
THE DOMAIN IN RANGE OF F OF X AND G OF X. NOTICE THAT WE HAVE A VERTICAL
ASYMPTOTE HERE AT X=0, WHICH MEANS THE DOMAIN WILL
NOT INCLUDE ZERO FOR F OF X. SO WE’LL HAVE
THE OPEN INTERVAL FROM NEGATIVE INFINITY TO ZERO AND THE OPEN INTERVAL
FROM ZERO TO INFINITY, AND THEN FOR THE RANGE NOTICE HOW IT WILL NOT INCLUDE
THE Y VALUE OF -1. SO WE’LL HAVE
THE OPEN INTERVAL FROM NEGATIVE INFINITY TO -1 AND THE OPEN INTERVAL
FROM -1 TO INFINITY. AND THEN FOR FUNCTION G OF X, NOTICE HOW THE DOMAIN
WILL ALWAYS BE WHEN X IS GREATER THAN ZERO, SO WE’LL HAVE THE OPEN
INTERVAL FROM ZERO TO INFINITY AND THEN FOR THE RANGE, NOTICE HOW THE GRAPH GOES
UP AND DOWN FOREVER WITH NO BREAKS, THEREFORE THE RANGE
IS ALL REAL NUMBERS OR FROM NEGATIVE INFINITY
TO POSITIVE INFINITY. NOW LET’S CONSIDER THE DOMAIN
OF OUR COMPOSITE FUNCTIONS. THE DOMAIN OF F OF G OF X, MUST CONTAIN THE RESTRICTIONS
OF THE DOMAIN OF THE INNER FUNCTION G OF X AND THE RESTRICTION
OF THE DOMAIN OF G OF X SUCH THAT THE RANGE OF G OF X IS IN THE DOMAIN
OF THE OUTER FUNCTION F OF X. SO WHAT THIS IS REALLY
TELLING US IS WE START WITH THE DOMAIN
OF THE INNER FUNCTION AND THEN BECAUSE THE OUTPUT
OF THE INNER FUNCTION BECOMES THE INPUT
INTO THE OUTER FUNCTION WE MAY HAVE MORE RESTRICTIONS
ON THE DOMAIN OF OUR COMPOSITE FUNCTION IF THE OUTPUTS OF G,
THE INNER FUNCTION ARE NOT POSSIBLE INPUTS
IN TO F OR THE OUTER FUNCTION. SO FOR THE DOMAIN
OF F OF G OF X, WE’LL START
WITH THE RESTRICTIONS FOR THE DOMAIN OF G OF X
WHICH IS THIS INTERVAL HERE AND NOW WE ALSO HAVE
TO COMPARE THE OUTPUTS OF G TO THE POSSIBLE INPUTS INTO F,
THE OUTER FUNCTION. AND WE DO HAVE
A SLIGHT PROBLEM BECAUSE NOTICE HOW ZERO
IS NOT A POSSIBLE INPUT INTO FUNCTION F, BUT ZERO IS OUTPUT
FROM THE INNER FUNCTION G. SO WE HAVE TO EXCLUDE
THE X VALUE OF G THAT WOULD PRODUCE AN OUTPUT
OF ZERO FOR FUNCTION G. AND IF WE CONSIDER
THIS POINT HERE WHERE THE X COORDINATE IS 1
AND THE Y COORDINATE IS ZERO THIS IS TELLING US
THAT WHEN G HAS AN INPUT OF 1 THE OUTPUT IS ZERO BUT SINCE ZERO
CANNOT BE AN INPUT INTO THE OUTER FUNCTION F WE MUST ALSO EXCLUDE 1 FROM THE DOMAIN
OF OUR COMPOSITE FUNCTION. SO THE DOMAIN OF F OF G OF X
WILL BE THIS INTERVAL HERE FROM ZERO TO INFINITY BUT WE ALSO HAVE TO EXCLUDE
X=1. WHICH MEANS THE DOMAIN
OF OUR COMPOSITE FUNCTION WILL BE THE OPEN INTERVAL
FROM ZERO TO 1 AND THE OPEN INTERVAL
FROM 1 TO INFINITY. NOW LET’S CONSIDER
THE DOMAIN OF G OF X OF X. AGAIN, WE’LL START
WITH THE RESTRICTIONS OF THE DOMAIN OF F, WHICH IS GIVEN
BY THIS INTERVAL HERE BUT NOW WE HAVE TO COMPARE
THE OUTPUTS OF F TO THE INPUTS OF G, AND NOTICE HOW WE HAVE
ANOTHER ISSUE BECAUSE NOTICE
HOW ALL THE OUTPUTS OF F ARE NOT POSSIBLE INPUTS
INTO THE OUTER FUNCTION G. SO WE NEED TO MAKE
MORE RESTRICTIONS ON THIS YELLOW INTERVAL HERE, SO THAT THE OUTPUTS WILL
ONLY BE FROM ZERO TO INFINITY. SO LET’S TAKE A LOOK
AT THE GRAPH OF F, WE WANT TO DETERMINE
THE INTERVAL FOR WHICH THE OUTPUTS OF F
ARE FROM ZERO TO INFINITY, WHICH WOULD BE THIS PIECE
OF THE GRAPH HERE. SO WE HAVE TO EXCLUDE
THE VALUES WHEN X IS GREATER THAN
OR EQUAL TO 1 AND WHEN X IS LESS THAN ZERO BECAUSE THESE X VALUES PRODUCE
OUTPUTS OF F THAT ARE NOT IN THE DOMAIN
OF THE OUTER FUNCTION G. WHICH MEANS THAT THE DOMAIN
OF OUR COMPOSITE FUNCTION G OF F OF X IS GOING TO BE THE
OPEN INTERVAL FROM ZERO TO 1. THESE ARE THE ONLY INPUTS
FOR THE INNER FUNCTION THAT PRODUCE OUTPUTS THAT ARE POSSIBLE INPUTS
INTO FUNCTION G. SO AS YOU CAN SEE,
DETERMINING THE DOMAIN OF COMPOSITE FUNCTIONS
GRAPHICALLY CAN SOMETIMES BE QUITE
CHALLENGING, BUT I HOPE YOU FOUND
THESE TWO EXAMPLES HELPFUL.  

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