Ex 1:  Determine the Domain and Range of the Graph of a Function
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Ex 1: 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. A DOMAIN IS A SET
OF ALL POSSIBLE X VALUES FOR THE FUNCTION,
AND THE RANGE IS A SET OF ALL POSSIBLE
Y VALUES FOR THE FUNCTION. WELL, X VALUES ARE ALONG
THE HORIZONTAL AXIS, AND Y VALUES ARE ALONG
THE VERTICAL AXIS. SO TO HELP US DETERMINE THE
DOMAIN OF THIS FUNCTION, WE WANT TO PROJECT THIS FUNCTION
ONTO THE X AXIS OR ANALYZE IT TO DETERMINE HOW THE GRAPH
BEHAVES FROM LEFT TO RIGHT. SO IF WE WERE TO PROJECT
THIS FUNCTION ON TO THE X AXIS, OR IF WE HAD A BRIGHT LIGHT
UP HERE THAT WAS SHINING DOWN ON THE FUNCTION, IT WOULD
PRODUCE A SHADOW ON THE X AXIS FROM -2 TO +4. BUT NOTICE HOW WE HAVE
AN OPEN POINT HERE AT X=-2, SO IT WOULD NOT INCLUDE
THE VALUE OF -2. SO WE’LL MAKE
AN OPEN POINT THERE. BUT BECAUSE THIS POINT
IS CLOSED, IT WOULD INCLUDE +4. SO THE DOMAIN OF THIS FUNCTION
WOULD BE THE INTERVAL FROM -2 TO +4 NOT INCLUDING -2,
BUT INCLUDING +4. AND WE CAN EXPRESS
THIS A COUPLE WAYS. USING INEQUALITIES WE CAN SAY
X IS GREATER THAN -2 AND LESS THAN OR=TO 4. OR IF WE WANT TO EXPRESS
THIS USING INTERVAL NOTATION, IT WOULD BE THE INTERVAL
FROM -2 TO +4 WHERE IT’S CLOSED ON +4 OR INCLUDES +4,
SO WE MAKE A BRACKET. AND IT’S OPEN ON -2, SO WE USE
A ROUNDED PARENTHESIS HERE. NOW FOR THE RANGE, WE’RE GOING
TO DO THE SAME TYPE OF ANALYSIS BUT NOW WE WANT TO PROJECT
THE FUNCTION ONTO THE Y AXIS OR ANALYZE THE GRAPH
TO DETERMINE HOW IT BEHAVES VERTICALLY. THE LOWEST POINT ON THIS GRAPH
WOULD BE RIGHT HERE AT Y=0. AND THEN NOTICE HOW IT EXTEND
ALL THE WAY UP TO Y=+8. AND IN THIS CASE
IT DOES INCLUDE 0 AND IT ALSO INCLUDES +8,
SO THE RANGE WOULD BE THE CLOSED INTERVAL FROM 0 TO 8. AGAIN, WE’RE GOING TO EXPRESS
THIS USING INEQUALITIES AS Y IS GREATER THAN OR=TO 0
AND LESS THAN OR=TO 8. OR USING INTERVAL NOTATION,
WE HAVE THE INTERVAL FROM 0 TO 8,
AND IT’S CLOSED BOTH ON 0 AND 8
BECAUSE IT WOULD INCLUDE THE ENDPOINTS. NOW LET’S TAKE A LOOK
AT ANOTHER EXAMPLE. AGAIN, WE’LL FIRST TRY
TO DETERMINE THE DOMAIN, WHICH IS THE SET
OF ALL POSSIBLE X VALUES, AND X VALUES ARE ALONG
THE HORIZONTAL AXIS. SO WE WANT TO PROJECT
THIS FUNCTION ONTO THE X AXIS OR ANALYZE IT TO DETERMINE
HOW IT BEHAVES MOVING FROM LEFT TO RIGHT. WELL, THE LEFT MOST POINT
ON THIS GRAPH WOULD BE RIGHT HERE AT X=-2,
AND THEN IT WOULD INCLUDE EVERY X VALUE
ALL THE WAY OUT TO +2. BUT NOTICE HOW WE HAVE
AN OPEN POINT RIGHT HERE AT X=2, SO IT DOES
NOT INCLUDE +2, SO IT’S AN OPEN POINT HERE. IT DOES INCLUDE THIS POINT HERE,
SO IT’S CLOSED HERE. SO THE DOMAIN
WOULD BE THE INTERVAL FROM -2 TO 2 INCLUDING -2,
AND NOT INCLUDING 2. SO WE COULD SAY
X IS GREATER THAN OR=TO -2 AND LESS THAN +2,
OR USING INTERVAL NOTATION, WE HAVE THE INTERVAL
FROM -2 TO 2, CLOSED ON -2 AND OPEN ON 2. NOW, TO DETERMINE THE RANGE,
WE WANT TO PROJECT THE FUNCTION ONTO THE Y AXIS
OR THE VERTICAL AXIS, OR ANALYZE
THE FUNCTION TO SEE HOW IT BEHAVES VERTICALLY. THE LOWEST POINT ON THIS GRAPH
WOULD BE RIGHT HERE AT Y=-4. NOTICE HOW IT WOULD INCLUDE -4
BECAUSE THIS POINT IS CLOSED. AND THEN THE GRAPH
INCLUDES EVERY Y VALUE ALL THE WAY UP TO +4,
NOT INCLUDING +4 AGAIN BECAUSE
OF THIS OPEN POINT. SO THE RANGE WOULD BE
THE INTERVAL FROM -4 TO 4, INCLUDING -4
BUT NOT INCLUDING +4. SO WE’D HAVE
Y IS GREATER THAN OR=TO -4 AND LESS THAN 4. USING INEQUALITIES OR USING
INTERVAL NOTATION FROM -4 TO 4, OPEN ON 4, AND CLOSED ON -4. OKAY. I HOPE YOU FOUND
THESE TWO EXAMPLES HELPFUL. WE’LL TAKE A LOOK
AT TWO MORE IN THE NEXT VIDEO.  

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