 – WHEN THE DOMAIN AND THE RANGE
OF A FUNCTION CAN OFTEN BE DONE IN MORE THAN ONE WAY AND THE BEST WAY IS REALLY
BASED UPON HOW MUCH YOU KNOW ABOUT A GIVEN FUNCTION, REGARDLESS OF THIS
THE GOAL IS GIVEN A FUNCTION WE NEED TO BE ABLE TO FIND
THE DOMAIN AND THE RANGE. LET’S GO AHEAD AND REVIEW
WHAT DOMAIN AND RANGE ARE. THE DOMAIN OF A FUNCTION
IS A SET OF ALL X VALUES OR INPUTS OF THE POINTS
ON THE GRAPH. YOU CAN THINK OF THE DOMAIN AS
A CURVED SHADOW ONTO THE X-AXIS OR HOW IT BEHAVES
FROM LEFT TO RIGHT. LET’S TAKE A LOOK AT THIS
DOWN BELOW. IF I ASK YOU HOW THIS GRAPH
BEHAVED ONLY FROM LEFT TO RIGHT OR IF I TRIED TO SHADOW THIS
ONTO THE X-AXIS, I MIGHT ASK YOU,
“WHAT IS THE LEFTMOST POINT?” AND YOU CAN SEE THIS POINT HERE
IS THE LEFTMOST POINT. IF I PROJECT THIS
ONTO THE X-AXIS, IT WOULD END UP RIGHT HERE. AND IF I ASKED YOU
THE SAME THING ABOUT THE RIGHTMOST POINT
ON THE GRAPH, THAT WOULD BE HERE. PROJECTING IT ONTO THE X-AXIS,
IT WOULD LAND HERE. AND OF COURSE, ALL OF THESE
OTHER POINTS ON THE FUNCTION WOULD BE PROJECTED ALONG
THIS INTERVAL ON THE X-AXIS, AND IN FACT THIS INTERVAL
FROM 0 TO 3 IS OUR DOMAIN. THE RANGE IS DONE
IN A SIMILAR FASHION, EXCEPT NOW WE’RE GOING TO BE
CONCERNED ABOUT Y VALUES. THE RANGE OF A FUNCTION
IS THE SET OF ALL Y VALUES OR OUTPUTS OF THE POINTS
ON THE GRAPH. THE RANGE CAN BE VIEWED AS THE
CURVED SHADOW ONTO THE Y-AXIS OR HOW IT BEHAVES
MOVING UP AND DOWN. IF I PROJECT THIS CURVE
ONTO THE Y-AXIS, I MIGHT ASK YOU, “WHAT IS
THE LOWEST POINT ON THE GRAPH?” AGAIN, THAT WOULD BE HERE, BUT NOW I’LL PROJECT IN ONTO
THE Y-AXIS OVER HERE TO -2. THE HIGHEST POINT ON THE GRAPH
IS ACTUALLY 2 OF THEM, BUT THEY WOULD BOTH PROJECT
OR SHADOW ONTO THE Y-AXIS HERE AT +1. ALL THESE OTHER POINTS
WOULD SHADOW SOMEWHERE ON THIS INTERVAL
IN BETWEEN THESE 2 VALUES, AND IN FACT THE RANGE OF THIS
FUNCTION IS FROM -2 TO +1 AS WE’VE SHOWN HERE IN GREEN. LET’S TAKE A LOOK
AT A COUPLE OTHER EXAMPLES. LET’S TAKE THE DOMAIN AND THE
RANGE OF THE FOLLOWING RELATION. OKAY, THE DOMAIN AGAIN
IS THE SET OF ALL X VALUES, SO WE’RE GOING TO LIST
ALL OF THE X COORDINATES OF THESE POINTS. NOW IF THE VALUE OCCURS
MORE THAN ONCE, WE WILL ONLY LIST IT ONCE. SO WE HAVE (2,4)(3,6)(2,3)(4,6),
THAT’S OUR DOMAIN. OUR RANGE WOULD BE THE SET
OF Y COORDINATES IN THIS CASE, SO WE HAVE A (-3,6)(-1,6) AND 3. AGAIN IF IT OCCURS MORE THAN
ONCE, WE ONLY LIST IT ONCE, AT -3, -1, 3 AND 6. NOTICE HOW I LISTED THEM
IN ORDER FROM LEAST TO GREATEST, THAT’S OFTEN WHAT IS DONE. NEXT QUESTION,
“IS THIS RELATION A FUNCTION?” AND THE QUESTION OCCURS, REMEMBER EVERY X CAN ONLY HAVE
1 Y AND NOTICE THAT THE X VALUE
OF + 2 IS PAIRED WITH 2 DIFFERENT
Y VALUES, THEREFORE IT’S NOT A FUNCTION. REMEMBER IN ORDER FOR A RELATION
TO BE A FUNCTION, EVERY X VALUE CAN ONLY HAVE
1 Y VALUE. LET’S TAKE A LOOK AT FINDING
THE DOMAIN AND THE RANGE OF A GIVEN FUNCTION AND F OF X=THE ABSOLUTE VALUE
OF -1. IT’S OFTEN EASIER TO FIND THE DOMAIN AND THE RANGE
OF A FUNCTION IF YOU LOOK AT ITS GRAPH. NOW IF I PROJECT THIS ONTO
THE X-AXIS TO FIND THE DOMAIN, ONE THING YOU NOTICE
IS THAT THIS FUNCTION DOES CONTINUE FOREVER
IN BOTH DIRECTIONS. IF I TRIED TO PROJECT
THIS FUNCTION ONTO THE X-AXIS, I WOULD END UP HAVING A SHADOW
ON THE ENTIRE X-AXIS, THEREFORE THE DOMAIN
OF THIS FUNCTION WOULD BE ALL REAL NUMBERS. USING INTERVAL NOTATION,
IT WOULD LOOK LIKE THIS. NOW IF I TRIED TO PROJECT THIS
ONTO THE Y-AXIS OR ASK HOW THIS GRAPH
BEHAVES GOING UP AND DOWN ALONG THE Y-AXIS, AGAIN, THIS GRAPH I PROJECT,
IT GOES UP FOREVER, SO I WOULD END UP PROJECTING
ALL OF THE POINTS ON THE Y-AXIS UNTIL I GOT DOWN TO HERE. LOOKS LIKE THIS AT -1, SO THE RANGE FOR THIS FUNCTION
WOULD BE FROM -1 TO +INFINITY. ANOTHER WAY TO THINK OF THIS,
ABOUT THE ABSOLUTE VALUE OF X, YOU COULD PROBABLY FIGURE
ALL OF THIS OUT JUST BY USING SOME REASONING, MEANING IF YOU TAKE
THE ABSOLUTE VALUE OF X, ARE THERE ANY RESTRICTIONS ON X? THE ANSWER IS NO. YOU CAN TAKE THE ABSOLUTE VALUE
OF ANY REAL NUMBER, THEREFORE THAT’S WHY THE DOMAIN
IS ALL REAL NUMBERS. HOWEVER, LET YOU DO KNOW
THAT THE ABSOLUTE VALUE OF X WILL ONLY YIELD A VALUE
THAT IS 0 OR LARGER. AND IF YOU TAKE A NUMBER THAT IS
0 OR LARGER AND SUBTRACT 1, THE ONLY POSSIBLE OUTPUTS
WOULD BE -1 THROUGH +INFINITY. LET’S TAKE A LOOK
AT THE DOMAIN AND RANGE OF A RATIONAL FUNCTION. I’M ACTUALLY GOING TO START
WITH THE ALGEBRAIC METHOD FIRST. IF YOU LOOK AT THE DOMAIN,
WE’RE LOOKING AT X VALUES. AND REALLY THE QUESTION IS, ARE THERE ANY RESTRICTIONS ON
WHAT X CAN BE IN THIS FUNCTION? AND IN FACT, WE KNOW THAT
WE CAN’T HAVE DENOMINATOR OF 0. THEREFORE IF I FIND THE VALUE OF
X THAT MAKES THE DENOMINATOR 0, I CAN EXCLUDE THIS
FROM THE DOMAIN. THE ONLY VALUE THAT X
COULD NOT BE WOULD BE 1.5 BECAUSE IF I USE THE VALUE
OF 1.5 OR 3/2, IT WOULD MAKE THE DENOMINATOR 0 AND THAT IS THE ONLY X VALUE THAT WOULD MAKE THE DENOMINATOR
0. THEREFORE OUR DOMAIN
WOULD BE ALL REALS, EXCEPT 1.5. AND USING INTERVAL NOTATION, WE
WOULD EXPRESS IT IN THIS MANNER. NOW IN ORDER TO DETERMINE
THE RANGE ALGEBRAICALLY, THIS IS A BIT CHALLENGING BUT YOU HAVE TO KIND OF
REASON THIS OUT BASED UPON WHAT YOU KNOW
ABOUT HOW FRACTIONS BEHAVE. WHAT’S GOING TO HAPPEN
TO THE VALUE OF F OF X AS X GETS REALLY, REALLY LARGE? — IS GOING TO STAY AT 5. THE DENOMINATOR IS GOING TO GET
LARGER AND LARGER. SO IF I HAVE A FRACTION
WITH A FIXED NUMERATOR AND AN INCREASING DENOMINATOR, IT’S ACTUALLY
GOING TO APPROACH 0. WILL NEVER GET THERE,
BUT IT WILL APPROACH 0. THEREFORE THE RANGE WOULD BE
ALL REAL NUMBERS, EXCEPT 0. IT WOULD APPROACH 0, BUT IT
WOULD ACTUALLY NEVER GET THERE. THE ACTUAL DOMAIN AND RANGE
AND COMPARE IT TO THE GRAPH, REMEMBER WE SAID THAT IN ORDER TO FIND THE DOMAIN
OF A FUNCTION, YOU PROJECT IT ALONG THE X-AXIS OR ASK HOW IT BEHAVES
FROM LEFT TO RIGHT. NOTICE HOW ALL THESE POINTS HERE
WOULD BE PROJECTED ONTO THE X-AXIS, BUT RIGHT IN HERE
THERE’S A LITTLE BIT OF A BREAK WHERE THERE IS NOT A VALUE
TO PROJECT ONTO THE X-AXIS AND THEN IT WOULD PICK UP AND
PROJECT ON THE X-AXIS THIS WAY. AND IN FACT WHAT WE HAVE FOUND
IS THIS VALUE THAT’S MISSING WOULD BE THAT 1.5, BECAUSE THAT’S WHAT MAKES
THE DENOMINATOR 0. NOW, A SIMILAR THING HAPPENS WHEN YOU PROJECT THIS
ONTO THE Y-AXIS, BECAUSE ALL OF THESE POINTS
WOULD BE PROJECTED ONTO THE Y-AXIS AND AGAIN THERE’S BREAK
RIGHT HERE AT Y=0, AND THEN IT WOULD BE PROJECTED
AGAIN ONTO THE Y-AXIS. SO ALL VALUES EXCEPT 0 WOULD BE
IN OUR RANGE AS WE FOUND HERE. NOW, THIS IS QUITE TRICKY AND IT’S GOING TO TAKE
SOME PRACTICE, BUT I HOPE THAT HELPS. LET’S TAKE A LOOK AT ONE MORE. LET’S TAKE A LOOK AT THE
ALGEBRAIC METHOD AGAIN FIRST. WHAT WE KNOW ABOUT THE ABSOLUTE
VALUE OF SOME ARGUMENT IS THAT THE RADICAND
OR THE NUMBER UNDERNEATH HAS TO BE GREATER THAN
OR EQUAL TO 0. SO 4X + 2 MUST BE GREATER THAN
OR EQUAL TO 0, AND THIS WOULD GIVE US
OUR DOMAIN. SO IF I SUBTRACT 2 DIVIDED BY 4, WE’RE GOING TO GET X IS GREATER
THAN OR EQUAL TO -1/2. AND IN FACT IF YOU TAKE A LOOK
AT THE GRAPH HERE, YOU CAN SEE, IT’S KIND OF HARD TO READ, BUT IT DOES START RIGHT HERE
AT -1/2. PROJECTING IT ONTO THE X-AXIS, AGAIN YOU WOULD HAVE THE DOMAIN
LOOKING SOMETHING LIKE THIS. SO OUR DOMAIN IS GOING TO START
AT -1/2 AND GO TO INFINITY. OKAY. NOW, IF WE TAKE A LOOK
AT THE RANGE, KIND OF AT AN ALGEBRAIC METHOD, AGAIN, WELL YOU KNOW WHEN YOU
TAKE THE PRINCIPLE SQUARE ROOT, THE SMALLEST VALUE YOU CAN GET
IS 0 AND EVERYTHING ELSE
HAS TO BE POSITIVE. SO IN FACT, OUR RANGE
WOULD BE FROM 0 TO +INFINITY. RELATING THIS TO OUR GRAPH IF
I PROJECT THIS ONTO THE Y-AXIS, ALL OF THESE POINTS
WOULD PROJECT OVER TO HERE. NOW, THIS GRAPH DOES CONTINUE
TO GO UP FOREVER. AND THE QUESTION IS,
HOW LOW DOES IT GO? IT GOES AS LOW
AS THIS POINT HERE WHICH, AGAIN, WOULD BE AT 0
ON THE Y-AXIS, AND THERE’S OUR RANGE IN RED.