Ex:  Determine the Domain of an Absolute Value Function
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Ex: Determine the Domain of an Absolute Value Function


– WE WANT TO DETERMINE
THE DOMAIN AND RANGE OF THE FUNCTION F(X)=
THE ABSOLUTE VALUE OF X + 2 – 3. THE DOMAIN CONSISTS
OF ALL POSSIBLE INPUTS OR X VALUES FOR THE FUNCTION
AND THE RANGE CONSISTS OF ALL POSSIBLE OUTPUTS
OR Y VALUES FOR THE FUNCTION. NOW THERE ARE A COUPLE
OF WAYS OF DETERMINING THE DOMAIN AND RANGE. WE COULD USE WHAT WE KNOW
ABOUT THE PROPERTIES OF THE ABSOLUTE VALUE FUNCTION
TO DETERMINE THE DOMAIN AND RANGE OR WE COULD
ALSO MAKE A GRAPH OF THE FUNCTION
AND ANALYZE THE GRAPH. LET’S START
BY ANALYZING THE GRAPH. SO THE FIRST STEP
WOULD BE TO EITHER MAKE A T-TABLE OR USE TECHNOLOGY
TO CREATE THE GRAPH PROVIDED HERE. ONCE WE HAVE THE GRAPH,
WE CAN ANALYZE THE BEHAVIOR OF THE FUNCTION HORIZONTALLY
AND VERTICALLY TO HELP DETERMINE THE DOMAIN AND THE RANGE. WHAT I MEAN BY THAT IS,
TO DETERMINE THE DOMAIN, IF WE WERE TO PROJECT
THIS FUNCTION ONTO THE X-AXIS, NOTICE HOW IT INCLUDES
THE ENTIRE X-AXIS. ANOTHER WAY TO THINK OF THIS IS,
THE GRAPH CONTINUES TO THE RIGHT AND CONTINUES TO THE LEFT
INDEFINITELY WITHOUT ANY HOLES OR BREAKS. SO THE DOMAIN APPROACH
IS POSITIVE INFINITY TO THE RIGHT
AND NEGATIVE INFINITY TO THE LEFT. THEREFORE, THE DOMAIN
OF THIS FUNCTION WOULD BE ALL REAL NUMBERS. OR IF WE WANTED
TO USE INTERVAL NOTATION, WE COULD USE THE INTERVAL
FROM NEGATIVE INFINITY TO POSITIVE INFINITY. NOW TO DETERMINE THE RANGE,
WE’D HAVE TO PROJECT THE GRAPH ONTO THE Y-AXIS
OR DETERMINE HOW THE GRAPH BEHAVES VERTICALLY. NOTICE HOW THE LOWEST POINT
ON THE GRAPH IS THIS POINT HERE WITH A Y VALUE OF -3. AND THEN FROM HERE, OF COURSE,
THE GRAPH IS MOVING TO THE RIGHT AND LEFT VERY QUICKLY,
BUT NOTICE HOW IT’S ALSO MOVING UPWARD. SO IF WE PROJECTED THIS GRAPH
ONTO THE Y-AXIS, IT WOULD ACTUALLY APPROACH
POSITIVE INFINITY ALONG THE Y-AXIS,
WHICH MEANS THE RANGE WOULD BE FROM -3
TO POSITIVE INFINITY AND IT DOES INCLUDE -3. SO WE CAN SAY THE RANGE
IS GREATER THAN OR EQUAL TO -3. OR USING INTERVAL NOTATION,
IT WOULD BE THE INTERVAL FROM -3 TO INFINITY
AND IT’S CLOSED ON -3 BECAUSE IT DOES INCLUDE -3. NOW IF WE DIDN’T WANT TO MAKE
A GRAPH AND WE KNEW A LOT ABOUT THE PROPERTIES
OF THE ABSOLUTE VALUE FUNCTION, WE COULD USE REASONING
TO DETERMINE THE DOMAIN AND RANGE. WHAT I MEAN BY THAT IS,
WE COULD THINK TO OURSELVES WHAT ARE THE POSSIBLE VALUES
OF X THAT WE COULD USE TO SUBSTITUTE IN HERE
AND THEN EVALUATE THE FUNCTION? AND THE ANSWER IS,
WE COULD USE ANY REAL NUMBER HERE,
ADD 2 TO IT, TAKE THE ABSOLUTE VALUE
AND THEN SUBTRACT 3. AND THAT’S THE REASON WHY
OUR DOMAIN IS ALL REAL NUMBERS. AND THEN FOR THE RANGE,
IF WE KNEW THE ABSOLUTE VALUE OF A NUMBER IS ALWAYS
GOING TO BE>THAN OR=TO 0, WELL,
THIS IS ALWAYS>THAN OR=TO 0
AND THEN WE SUBTRACT 3, THEN THE FUNCTION VALUE,
OR Y VALUE, IS ALWAYS GOING TO BE>THAN
OR=TO -3. AGAIN, IF THIS VALUE RIGHT HERE
IS ALWAYS GOING TO BE 0 OR SOME POSITIVE NUMBER,
THEN THE FUNCTION VALUE. OR Y VALUE, MUST ALWAYS
BE>THAN OR=TO -3 WHICH IS OUR RANGE. SO I KNOW IT CAN BE CHALLENGING
TO JUST THINK ABOUT WHAT THE DOMAIN AND RANGE
WOULD BE, GIVEN THE FUNCTION. SO I THINK IT’S ALWAYS HELPFUL
TO SKETCH A GRAPH OF THE FUNCTION
AND ANALYZE THE GRAPH, IF NOTHING ELSE,
JUST TO VERIFY THE DOMAIN AND RANGE.  

8 Comments

  • Carl Mirita

    Awesome! Better then my teacher! It always seems like teachers do worse when they have a class then over the internet. It's probably because their more relaxed or something.

  • SummativeWritting

    Well, the reason I feel this video is bad is because it took "bullcleo1" 3:45 to go over perhaps the easiest example of an absolute value function. The only person this would help is someone who didn't know the definition of the domain and/or range or perhaps someone who doesn't know how to graph an absolute value function. I think the person who this video helps is bad at math. 😛

  • Allyson

    Anyone who says that this video was "bad" or "stupid" needs to, in my opinion, keep their mouths shut. Not everyone can look at a math problem and learn how to do it in a mere moment. For some, it takes "Oh, right! Now I remember!" to actually get a new lesson. It is VERY rude to insult people on the way, or pace, at which they learn. If you can remember it and don't need any reminding or pointers, good for you, as for making the assumption that everyone else who needs a little reminding is "bad at math", well, that is strictly impossible to prove true, and it is a very not-well-thought-out remark. Keep your opinion-based, rude comments to yourself.

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