Ex:  The Domain of Rational Functions
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Ex: The Domain of Rational Functions


– WE WANT TO DETERMINE
THE DOMAIN OF THE FOLLOWING
RATIONAL FUNCTIONS. TO DETERMINE THE DOMAIN
OF A RATIONAL FUNCTION, WE WANT TO FIND THE X VALUES THAT WE MUST EXCLUDE
FROM THE DOMAIN AND SINCE A FRACTION BAR
REPRESENTS DIVISION AND WE KNOW THAT DIVISION
BY ZERO IS UNDEFINED, TO FIND THE DOMAIN
OF THE RATIONAL FUNCTION, WE WANT TO SET THE DENOMINATOR
EQUAL TO ZERO AND SOLVE FOR X. THIS WILL TELL US WHICH VALUES
WE MUST EXCLUDE FROM THE DOMAIN. SO, FOR OUR FIRST FUNCTION, WE WANT TO SET 2X – 5=0
AND SOLVE FOR X. SO, WE’D ADD 5
TO BOTH SIDES OF THE EQUATION. THAT WOULD GIVE US 2X=5 AND THEN DIVIDE BOTH SIDES
BY 2. SO, THIS TELLS US THAT
X=5 1/2, WOULD MAKE THE DENOMINATOR
EQUAL TO ZERO, SO WE MUST EXCLUDE 5 1/2
FROM THE DOMAIN. SO, THE DOMAIN WOULD
BE ALL REAL NUMBERS OR WE CAN SAY ALL REALS EXCEPT
X=5 1/2. SO, IF WE WERE TO GRAPH
THE DOMAIN, WE WOULD GRAPH A NUMBER LINE,
PLOT 5 1/2. WE’D EXCLUDE THIS VALUE. SO, WE’D HAVE AN OPEN POINT HERE AND THEN WE’D SHADE EVERY OTHER
REAL NUMBER. SO, WE’D SHADE TO THE RIGHT
AND TO THE LEFT. SO, IF WE WANTED TO EXPRESS
THIS USING INTERVAL NOTATION, YOU’D BE APPROACHING POSITIVE
INFINITY ON THE RIGHT, NEGATIVE INFINITY ON THE LEFT. SO, USING INTERVAL NOTATION, WE’D HAVE THE INTERVAL FROM
NEGATIVE INFINITY TO 5 1/2, UNION 5 1/2
TO POSITIVE INFINITY. SO, HERE WE EXPRESS
A DOMAIN IN WORDS. HERE, WE EXPRESS IT AS A GRAPH AND HERE, WE EXPRESSED IT
USING INTERVAL NOTATION. FOR OUR SECOND FUNCTION, WE NEED
TO SET 2X SQUARED + 3X=0 AND SOLVE FOR X TO DETERMINE
THE EXCLUDED VALUES. SO, WE HAD THE EQUATION
2X SQUARED + 3X=0. FIRST STEP IN FACTORING
IS TO FACTOR OUT THE GREATEST COMMON FACTOR,
WHICH IS X. SO, I HAVE X x THE QUANTITY
2X + 3=0. SO, EITHER X=0 FROM THIS FIRST
FACTOR OR 2X + 3=0. HERE, WE WOULD SUBTRACT
3 AND DIVIDE BY 2. SO, WE’D HAVE X=-3 1/2. SO, WE MUST EXCLUDE ZERO
AND -3 1/2 FROM THE DOMAIN. SO, THE DOMAIN
WOULD BE ALL REAL NUMBERS. WE CAN ABBREVIATE REAL NUMBERS
BY USING THIS SYMBOL HERE, IT’S LIKE A TWO-LEGGED R, EXCEPT X=-3 1/2 AND 0. SO AGAIN, IF WE WANTED TO GRAPH
THE DOMAIN, WE WOULD SKETCH A NUMBER LINE AND EXCLUDE ZERO AND -3 1/2. SO, LET’S SAY ZERO IS HERE
AND -3 1/2 IS HERE, WE’D EXCLUDE THESE TWO VALUES AND GRAPH EVERY
OTHER REAL NUMBER. SO, WE’D GRAPH IN BETWEEN
TO THE RIGHT AND TO THE LEFT. SO, IF WE WANTED TO EXPRESS
THIS USING INTERVAL NOTATION, THIS WOULD BE NEGATIVE INFINITY; THIS WOULD BE POSITIVE INFINITY. SO, WE’D HAVE FROM NEGATIVE
INFINITY TO -3 1/2 UNION FROM -3 1/2 TO ZERO,
UNION FROM ZERO TO INFINITY. NOW FOR THE LAST EXAMPLE, WE TOOK THE DENOMINATOR
OF X SQUARED + 4X – 21=0 AND SOLVE FOR X TO DETERMINE
THE EXCLUDED VALUES. SO, WE’D HAVE
X SQUARED + 4X – 21=0. THIS DOES FACTOR. SO, THE FIRST TERMS WOULD BE THE
FACTOR OF X SQUARED, WHICH WOULD BE X AND X. NOW, WE WANT THE FACTORS OF -21 THAT ADD TO 4. THAT WOULD BE 7 AND -3. SO, THE SOLUTION TO THIS
EQUATION ARE X=-7 OR X=3, WHICH MEANS, THESE VALUES MAKE
THE DENOMINATOR EQUAL TO ZERO AND MUST BE EXCLUDED
FROM THE DOMAIN. SO, THE DOMAIN
WOULD BE ALL REAL NUMBERS OR ALL REALS EXCEPT
X=-7 AND 3. WE’LL GRAPH THE DOMAIN AS WELL. SO, WE WOULD PLOT -7, 3, EXCLUDE THESE VALUES
WITH OPEN POINTS AND GRAPH
EVERY OTHER REAL NUMBER. SO, WE’D GRAPH IN THE MIDDLE,
TO THE RIGHT, AND TO THE LEFT. SO, USING INTERVAL NOTATION, ON THE LEFT,
WE’D HAVE THE INTERVAL FROM NEGATIVE INFINITY
TO -7, UNION -7 TO 3, UNION 3 TO INFINITY. NOW, THERE IS ONE MORE THING
I’D LIKE TO MENTION, SOMETIMES RATIONAL FUNCTIONS
WILL SIMPLIFY BECAUSE THEY HAVE COMMON FACTORS BETWEEN THE NUMERATORS
AND DENOMINATORS, BUT WE DO NOT WANT
TO SIMPLIFY THEM BEFORE DETERMINING
THE DOMAIN AND RANGE. WHAT I MEAN BY THAT IS,
IF YOU TAKE A LOOK AT G OF X, WE COULD WRITE THIS AS G OF X
=THE NUMERATOR OF X AND THE DENOMINATOR
IN FACTORED FORM WOULD BE X x THE QUANTITY
2X + 3. SO, WE DO NOT WANT TO SIMPLIFY
OUT THIS COMMON FACTOR OF X BECAUSE THEN WE WOULD NOT
EXCLUDE ZERO FROM THE DOMAIN. SO, THIS COMMON FACTOR DOES
PRODUCE A HOLE IN THE FUNCTION, WHILE THE ZERO OF THIS FACTOR
PRODUCES A VERTICAL ASYMPTOTE SO IT DOES AFFECT THE GRAPH, BUT WE NEED ALL THE FACTORS
OF THE DENOMINATOR TO DETERMINE THE DOMAIN. OKAY, I HOPE
YOU FOUND THIS HELPFUL.  

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