Articles

Ex: Give the Domain and Range Given the Graph of a Function


– WE WANT TO DETERMINE
THE DOMAIN AND THE RANGE OF F(X) GIVEN
THE GRAPH OF F(X). WE ALSO WANT
TO WRITE THE ANSWER USING INTERVAL NOTATION
AND AS A COMPOUND INEQUALITY. THE DOMAIN OF A FUNCTION
IS A SET OF ALL POSSIBLE INPUTS OR X VALUES. SO TO DETERMINE THE DOMAIN
OF THE FUNCTION, WE’LL ANALYZE THE GRAPH FROM LEFT TO RIGHT
SINCE THE X VALUES RUN FROM LEFT TO RIGHT. AND THE RANGE OF A FUNCTION
IS A SET OF ALL POSSIBLE OUTPUTS OR Y VALUES,
ALSO THE FUNCTION VALUES. AND SINCE THE Y VALUES
RUN VERTICALLY, WE’LL ANALYZE THIS GRAPH
VERTICALLY TO DETERMINE THE RANGE OF THE FUNCTION. LET’S BEGIN BY DETERMINING
THE DOMAIN, WHICH WOULD BE THE SET OF ALL
POSSIBLE X VALUES. IF WE WERE TO PROJECT
THIS GRAPH UNDER THE X AXIS OR THINK HOW FAR LEFT
AND HOW FAR RIGHT THE GRAPH GOES,
NOTICE HOW IT STARTS AT -6. BECAUSE OF THE OPEN POINT,
IT DOES NOT INCLUDE -6. BUT THEN FROM THERE
IT GOES FROM -6 ALL THE WAY TO +4, BUT, AGAIN
BECAUSE OF THIS OPEN POINT HERE IT DOES NOT INCLUDE +4. SO THE DOMAIN OF THE FUNCTION
IS THE OPEN INTERVAL FROM -6 TO +4. SO USING INTERVAL NOTATION
WE WOULD HAVE A ROUNDED PARENTHESIS (-6,4). REMEMBER THESE ROUNDED
PARENTHESIS INDICATE THAT THE ENDPOINTS
ARE NOT INCLUDED. AND AS A COMPOUND INEQUALITY
WE CAN SAY THAT X IS GREATER THAN -6,
NOT GREATER THAN OR EQUAL TO BECAUSE, AGAIN,
IT DOES NOT INCLUDE -6, AND X IS ALSO LESS THAN 4. SO HERE’S A DOMAIN
USING INTERVAL NOTATION, AS WELL AS USING
A COMPOUND INEQUALITY. AND NOW WE’LL ANALYZE
THE GRAPH VERTICALLY TO DETERMINE THE RANGE. NOTICE THE LOWEST POINT
ON THE GRAPH WOULD BE THIS POINT HERE WHERE THE Y VALUE IS -5. BECAUSE THIS POINT IS CLOSED,
IT DOES INCLUDE THE VALUE OF Y=-5. SO THE RANGE STARTS
AT -5 INCLUDING -5. THERE ARE NO HOLES
AND BREAKS IN THE GRAPH, AND THE HIGHEST POINT
IN THE GRAPH WOULD BE THIS POINT HERE WHERE NOTICE
THAT Y IS +4. AGAIN, THIS IS A CLOSED POINT,
SO Y=4 IS IN THE RANGE, AND THEREFORE THE RANGE
IS THE CLOSED INTERVAL FROM -5 TO +4. SO USING INTERVAL NOTATION
WE WOULD NOW USE SQUARE BRACKETS TO MAKE SURE
THAT THE ENDPOINTS ARE INCLUDED. SO [-5,4]. AND NOW FOR
THE COMPOUND INEQUALITY, BECAUSE WE’RE TALKING
ABOUT THE RANGE WE’LL USE Y INSTEAD OF X. SO Y IS GREATER THAN
OR EQUAL TO -5 AND LESS THAN OR EQUAL TO +4. THIS WOULD BE THE DOMAIN
AND THE RANGE OF THE GIVEN FUNCTION. I HOPE YOU FOUND THIS HELPFUL.  

Leave a Reply

Your email address will not be published. Required fields are marked *