|
Question 84464: I do not fully understand what domain is and I am having trouble with the following problem:
y= square root of(9-x^2) + square root of(1-x^2)
The answer on my key is domain=[1,3] but I do not understand why. Can you please explain?
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! I do not fully understand what domain is and I am having trouble with the following problem:
y= square root of(9-x^2) + square root of(1-x^2)
The answer on my key is domain=[1,3] but I do not understand why. Can you please explain?
HERE X IS CALLED INDEPENDENT VARIABLE AND Y DEPENDENT VARIABLE.
OUR UNIVERSE IS REAL NUMBERS FOR THIS PURPOSE.
NOW , THIS DOES NOT MEAN THAT X CAN TAKE ANY VALUE IN REAL NUMBERS.
IT CAN TAKE ONLY SUCH VALUES WHICH MAKE THE GIVEN FUNCTION F(X) OR Y IN SHORT MEANINGFUL.THAT IS A MATHEMATICALLY ALLOWED OPERATION.
CERTAIN OPERATIONS LIKE DIVISION BY ZERO , SQUARE ROOT OF A NEGATIVE NUMBER,LOG OF ZERO OR NEGATIVE NUMBER ARE NOT DEFINED.EITHER THEY ARE CALLED INFINITY(PLUS OR MINUS)OR IMAGINARY.
SO WE SAY X CAN NOT TAKE SUCH VALUES WHICH REQUIRES US TO PERFORM SUCH OPERATIONS.
X CAN TAKE VALUES OF ALL OTHER REAL NUMBERS.THE SET OF THESE REAL NUMBERS IS CALLED DOMAIN OF THE FUNCTION.
IF YOU HAVE
Y=1/(X-3)...DOMAIN IS ALL REAL VALUES EXCEPT 3 AS IT WILL LEAD TO DIVISION BY ZERO .
IN YOUR EXAMPLE THE FIRST TERM BECOMES MEANING LESS IF
9-X^2<0.HENCE
9-X^2>=0
9>=X^2
X^2<=9
|X|<=3......THAT IS X SHOULD BE BETWEEN -3 AND +3
WE SHOW IT AS X = [-3,3]......[..]..BRACKETS INDICATE X CAN EQUAL THE BORDER VALUES
THAT IS X=3 AND X=-3 ARE ALLOWED.
NOW THERE IS A SECOND TERM.SQRT.(1-X^2)
SIMILAR ARGUMENT LEADS US TO CONCLUDE X = [-1,1]
THE COMBINATION OF THE I AND II CONDITIONS MAKE US TO CONCLUDE THAT X = [-1,1]
YOUR ANSWER IS [1,3]WHICH IS NOT CORRECT.
ON THE OTHER HAND,IF SECOND TERM IS SQ.RT(X^2-1)....THEN |X| >=|
X<=-1......OR............X>=1...COMBINING THIS WITH I CONDITION , WE GET
X=[1,3] AND [-3,-1]
|
|
|
| |
