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- 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.