Ex:  Determine the Domain of a Rational Function
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Ex: Determine the Domain of a Rational Function


– WE WANT TO DETERMINE
THE DOMAIN AND RANGE OF THE FUNCTION F(X)=1
DIVIDED BY THE QUANTITY X + 2. THE FIRST THING TO REMEMBER
IS THE DOMAIN CONSISTS OF ALL POSSIBLE
INPUTS OR X VALUES AND THE RANGE CONSISTS OF ALL
POSSIBLE OUTPUTS OR Y VALUES. NOW THERE’S A COUPLE
OF WAYS OF DETERMINING THE DOMAIN AND RANGE. ONE WAY WOULD BE
TO ANALYZE THE GRAPH TO DETERMINE THE DOMAIN
AND RANGE. THE OTHER WAY WOULD BE
TO USE REASONING BASED UPON WHAT WE KNOW ABOUT
THE PROPERTIES OF A RATIONAL FUNCTION. LET’S START
BY DETERMINING THE DOMAIN AND RANGE GRAPHICALLY. SO WHETHER WE USE TECHNOLOGY
OR A TABLE OF VALUES TO SKETCH THE GRAPH,
THE GRAPH OF THE FUNCTION WOULD LOOK LIKE THIS. ONE THING WE SHOULD
NOTICE RIGHT AWAY IS THAT X COULD NOT=-2
BECAUSE, IF IT DID, WE’D HAVE A 0
IN THE DENOMINATOR. AND AS A RESULT, WE HAVE
A BREAK IN THE GRAPH AND X=-2. THIS IS ACTUALLY
A VERY IMPORTANT ASPECT OF DETERMINING THE DOMAIN
OF THE GIVEN FUNCTION. SO TO DETERMINE THE DOMAIN,
SINCE THE X VALUES OCCUR ALONG THE X-AXIS,
WE WANT TO DETERMINE HOW THIS GRAPH
BEHAVES HORIZONTALLY. WELL, HORIZONTALLY,
IT’S EASY TO SEE THIS GRAPH AS APPROACHING
TO THE RIGHT AND APPROACHING
TO THE LEFT INDEFINITELY. BUT BECAUSE THERE’S A BREAK
IN THE GRAPH HERE, -2 IS NOT GOING TO BE
IN THE DOMAIN OF THE FUNCTION. ANOTHER WAY
THAT’S HELPFUL TO DETERMINE THE DOMAIN GRAPHICALLY
IS TO PROJECT THE FUNCTION ONTO THE X-AXIS. AND, AGAIN,
NOTICE IF WE DID THAT, THERE’D BE A BREAK HERE AT -2,
BUT THEN ALL THE X VALUES WOULD OCCUR ALONG THE X-AXIS
HERE AND HERE TO THE LEFT. SO, AGAIN, TO THE RIGHT,
WE’RE APPROACHING POSITIVE INFINITY
AND TO THE LEFT WE’RE APPROACHING
NEGATIVE INFINITY. SO THERE’S A COUPLE OF WAYS
TO EXPRESS THE DOMAIN. WE CAN SAY ALL REALS. THIS IS AN ABBREVIATION
FOR ALL REAL NUMBERS. SO ALL REALS EXCEPT X=-2. NOW WE COULD ALSO EXPRESS
THIS USING INTERVAL NOTATION. WE’D HAVE THE INTERVAL
FROM NEGATIVE INFINITY TO -2. NOW IT DOESN’T INCLUDE -2,
SO WE HAVE AN OPEN PARENTHESIS OR THE UNION FROM -2
TO POSITIVE INFINITY. THIS WOULD BE
THE DOMAIN EXPRESSED USING INTERVAL NOTATION. NOW TO DETERMINE
THE RANGE GRAPHICALLY, WE WANT TO DETERMINE
HOW THE GRAPH BEHAVES VERTICALLY UP INDEFINITELY
OR APPROACHES POSITIVE INFINITY ALONG THE Y-AXIS AND IT ALSO
MOVES DOWN INDEFINITELY APPROACHING NEGATIVE INFINITY
ALONG THE Y-AXIS. BUT THERE IS ALSO A BREAK
IN THE GRAPH AT Y=0. SO IF WE PROJECTED
THIS GRAPH ONTO THE Y-AXIS, THERE’D BE A BREAK
AT Y=0 HERE. THEN FROM 0,
IT WOULD APPROACH POSITIVE INFINITY MOVING UPWARD
AND NEGATIVE INFINITY MOVING DOWNWARD. SO WE CAN SAY THE RANGE
WOULD BE ALL REAL NUMBERS, OR ALL REALS, EXCEPT Y=0. OR WE CAN EXPRESS
THIS USING INTERVAL NOTATION. IT WOULD BE
FROM NEGATIVE INFINITY TO 0 UNION,
0 TO POSITIVE INFINITY. IF WE DIDN’T WANT TO USE UNION,
WE COULD ALSO USE OR. ANOTHER METHOD OF DETERMINING
THE DOMAIN AND RANGE WOULD BE TO JUST LOOK
AT THE FUNCTION AND USE WHAT WE KNOW
ABOUT THE PROPERTIES OF THE RATIONAL FUNCTION
TO DETERMINE THE DOMAIN AND RANGE. SO, FOR EXAMPLE,
ONE OF THE FIRST THINGS WE NOTICED IS THAT WE SAID
X COULD NOT=-2 BECAUSE WE’D HAVE DIVISION BY 0. SO KNOWING THAT X CAN’T=-2,
BUT COULD BE ANY OTHER VALUE, THAT WOULD LEAD
US TO THE DOMAIN, AS WE STATED HERE,
OR HERE USING INTERVAL NOTATION. NOW DETERMINING THE RANGE
WOULD BE A LITTLE MORE DIFFICULT JUST BY LOOKING AT THE FUNCTION. BUT NOTICE AS X APPROACHED -2,
THE DENOMINATOR IS GETTING SMALLER AND SMALLER. AND 1 DIVIDED BY
A VERY SMALL NUMBER IS ACTUALLY GOING TO APPROACH
EITHER POSITIVE INFINITY OR NEGATIVE INFINITY WHICH,
AGAIN, WE CAN SEE GRAPHICALLY. AS THE APPROACH -2
FROM THE RIGHT, NOTICE HOW THE Y VALUES
APPROACH POSITIVE INFINITY AND, IF WE APPROACH -2
FROM THE LEFT, NOTICE HOW WE WOULD BE
APPROACHING NEGATIVE INFINITY ALONG THE Y-AXIS. NOW THERE IS ONE MORE THING
WE SHOULD RECOGNIZE ABOUT THE RATIONAL FUNCTION
WHEN DETERMINING THE RANGE. THIS FRACTION IS NEVER
GOING TO BE=TO 0. WE CAN’T HAVE A FRACTION
=TO 0 IF THE NUMERATOR IS A CONSTANT. AND THAT’S THE REASON WHY
WE WOULD HAVE TO EXCLUDE 0 FROM THE RANGE OF THE FUNCTION,
WHICH LEADS US TO A RANGE OF ALL REALS EXCEPT
Y=0 OR HERE, OR HERE USING INTERVAL NOTATION. OKAY, I THINK WE’LL GO AHEAD
AND STOP HERE FOR THIS EXAMPLE. I HOPE THIS WAS HELPFUL.  

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