Ex 2:  Determine the Domain and Range of the Graph of a Function
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Ex 2: Determine the Domain and Range of the Graph of a Function


– WE WANT TO DETERMINE
THE DOMAIN AND RANGE OF A FUNCTION GIVEN
THE GRAPH OF THE FUNCTION. THE DOMAIN IS A SET
OF ALL POSSIBLE X VALUES OF THE FUNCTION. X VALUES OCCUR
ALONG THE X AXIS OR THE HORIZONTAL AXIS. AND THE RANGE IS A SET
OF ALL POSSIBLE Y VALUES OF THE FUNCTION,
AND Y VALUES OCCUR ALONG THE VERTICAL AXIS. SO IF WE’RE GIVEN
THE GRAPH OF A FUNCTION, AND WE WANT TO DETERMINE
THE DOMAIN OF THE FUNCTION, WE WANT TO PROJECT THE GRAPH
ONTO THE X AXIS, OR DETERMINE HOW THE GRAPH BEHAVES
HORIZONTALLY ALONG THE X AXIS. WHAT I MEANT BY THAT IS NOTICE
HOW THE LEFT MOST POINT OF THIS GRAPH OCCURS RIGHT HERE
WHEN X IS APPROACHING -3. AND THE RIGHT MOST POINT
ON THE GRAPH WOULD BE HERE WHEN X IS EQUAL TO +2. AND THE GRAPH WOULD ALSO CONTAIN
EVERY X VALUE BETWEEN -3 AND 2. BUT THERE’S
ONE MORE THING WE NEED TO BE CAREFUL
ABOUT HERE, -3 IS NOT GOING TO BE
IN THE DOMAIN OF THIS FUNCTION BECAUSE OF THIS OPEN POINT HERE. SO LETS MAKE AN OPEN POINT
HERE TO INDICATE THAT. BUT NOTICE THAT X=2,
THIS POINT IS CLOSED, SO IT WOULD INCLUDE +2. SO THE DOMAIN OF THIS FUNCTION
IS GOING TO BE FROM -3 TO +2, NOT INCLUDING -3
BUT INCLUDING +2. SO IF WE WANTED TO EXPRESS
THIS USING INEQUALITIES, WE WOULD SAY
X IS GREATER THAN -3 AND LESS THAN OR EQUAL TO +2. IF WE WANT TO USE
INTERVAL NOTATION, THE INTERVAL’S FROM -3 TO 2. IT INCLUDES 2
SO IT’S CLOSED ON 2, SO WE USE THIS SQUARE BRACKET. AND IT’S OPEN ON -3
BECAUSE IT DOES NOT INCLUDE -3, SO WE USE A ROUNDED PARENTHESIS. THESE TWO MEAN THE SAME THING. AND THEN TO DETERMINE THE RANGE,
WE NOW WANT TO PROJECT THIS FUNCTION
ONTO THE Y AXIS, OR DETERMINE
HOW IT BEHAVES VERTICALLY. SO, AGAIN, NOTICE
HOW THE LOWEST POINT ON THIS GRAPH HERE
IS APPROACHING -5, AND THEN IT INCLUDES
EVERY Y VALUE ALL THE WAY UP TO
THIS HIGH POINT WHEN Y IS +5. BUT NOTICE HOW IT’S NOT GOING
TO INCLUDE -5 BECAUSE OF THIS OPEN POINT,
BUT IT WILL INCLUDE +5 BECAUSE OF THIS CLOSED POINT. SO THE RANGE IS GOING
TO BE FROM -5 TO +5, NOT INCLUDING -5
AND INCLUDING 5. SO WE CAN SAY
Y IS GREATER THAN -5 AND LESS THAN OR EQUAL TO +5. OR USING INTERVAL NOTATION
SQUARE BRACKET FOR 5 BECAUSE IT INCLUDES 5,
AND A ROUNDED PARENTHESIS FOR -5 BECAUSE IT DOES NOT INCLUDE -5. NOW LET’S GO AND TAKE A LOOK
AT A SECOND EXAMPLE. WE’LL START BY DETERMINING
THE DOMAIN. SO WE WANT TO PROJECT
THIS FUNCTION ON TO THE X AXIS, OR DETERMINE
HOW IT BEHAVES HORIZONTALLY. WELL, THE LEFT MOST POINT OCCURS
RIGHT HERE AT X=-4, AND THEN NOTICE HOW THE GRAPH
MOVED TO THE RIGHT INDEFINITELY BECAUSE WE ARE ASSUMING
THIS GRAPH IS GOING TO CONTINUE IN THIS DIRECTION. SO THE DOMAIN
WOULD START AT -4 AND THEN MOVE
TO THE RIGHT INDEFINITELY, MEANING IT’S GOING TO APPROACH
POSITIVE INFINITY. SO THE DOMAIN
WOULD BE X IS GREATER THAN OR EQUAL TO -4,
OR USING INTERVAL NOTATION WE HAVE INTERVAL
FROM -4 TO INFINITY. IT INCLUDES -4
SO WE HAVE A BRACKET. AND THEN FOR INFINITY
WE ALWAYS USE A ROUNDED PARENTHESIS. AND THEN FOR THE RANGE,
WE WANT TO PROJECT THIS FUNCTION
ONTO THE Y AXIS, OR DETERMINE
HOW IT BEHAVES VERTICALLY. SO THE LOWEST POINT
ON THIS GRAPH IS RIGHT HERE AT Y=-4. THIS IS A CLOSED POINT
SO IT DOES INCLUDE -4. NOW, WHEN WE TRY TO DETERMINE
HOW HIGH THIS GRAPH GOES, WE NEED TO BE CAREFUL
BECAUSE OF COURSE IT IS MOVING TO THE RIGHT
VERY FAST, BUT NOTICE HOW IT ALSO IS MOVING UPWARD. SO EVEN THOUGH IT’S NOT SHOWING
ON THE SCREEN, THIS GRAPH WOULD CONTINUE
TO MOVE UPWARD, AND THEREFORE THE RANGE IS GOING TO APPROACH
POSITIVE INFINITY. SO THE RANGE
WOULD BE Y IS GREATER THAN OR EQUAL TO -4,
OR USING INTERVAL NOTATION, JUST LIKE FOR THE DOMAIN,
IT WOULD BE CLOSED ON -4 TO POSITIVE INFINITY. OKAY. SO I HOPE
THESE TWO EXAMPLES WERE HELPFUL.  

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