Ex: Domain and Range of Radical Functions

Ex: Domain and Range of Radical Functions


– WE WANT TO FIND THE DOMAIN AND
RANGE FOR EACH RADICAL FUNCTION. TO REVIEW, THE DOMAIN IS SET OF
ALL POSSIBLE INPUTS OR X VALUES, AND THE RANGE IS THE SET OF ALL
POSSIBLE OUTPUTS, WHICH WOULD BE THE Y VALUES
OR FUNCTION VALUES. NOTICE F OF X IS A SQUARE ROOT
FUNCTION, SO THE INDEX IS 2. G OF X IS A CUBE ROOT FUNCTION,
SO THE INDEX IS 3. AND H OF X IS THE — ROOT
FUNCTION, SO THE INDEX IS 4. DETERMINING WHETHER THE INDEX
IS ODD OR EVEN HELPS US DETERMINE
THE DOMAIN AND RANGE OF THE RADICAL FUNCTION. SO FOR EXAMPLE,
IF WE HAVE A RADICAL FUNCTION IN THE FORM F OF X=THE
PRINCIPAL NTH ROOT OF X, IF THE INDEX END IS EVEN THEN THE RADICAND
OR THE EXPRESSION UNDERNEATH THE SQUARE ROOT
HAS TO BE GREATER OR EQUAL TO 0, MEANING THE RADICAND
HAS TO BE NON NEGATIVE. AND IF THE RADICAND
IS NON NEGATIVE, THEN THE FUNCTIONS VALUES
OR THE Y VALUES WILL ALWAYS BE GREATER
OR EQUAL TO 0. BUT IF N IS ODD
THEN THE RADICAND OR X CAN BE ANY REAL NUMBER. AND IF THE RADICAND CAN BE
POSITIVE, NEGATIVE, OR ZERO THAT MEANS THE FUNCTION VALUES
WOULD ALSO BE ANY REAL NUMBER. SO GOING BACK TO OUR THREE
EXAMPLES, SINCE F OF X=THE PRINCIPAL
SQUARE ROOT OF 2 – X, OUR RADICAND 2 – X MUST BE
GREATER THAN OR EQUAL TO 0. SO IF WE SET THIS UP AS
INEQUALITY, WE WOULD HAVE 2 – X MUST BE GREATER THAN OR=0. AND NOW WE CAN SOLVE FOR X TO DETERMINE
THE DOMAIN OF THE FUNCTION. SO WE’LL SUBTRACT 2
ON BOTH SIDES, THAT WOULD GIVE US -X
WHICH GREATER THAN OR=-2. NOW, TO SOLVE FOR X,
WE’LL DIVIDE BOTH SIDES BY -1. BUT REMEMBER
WHEN SOLVING INEQUALITY, IF WE MULTIPLY OR DIVIDE
BY A NEGATIVE WE HAVE TO REVERSE
THE INEQUALITY SYMBOL. SO THIS WOULD GIVE US X
IS LESS THAN OR EQUAL TO 2, WHICH WOULD BE THE DOMAIN
OF THE GIVEN FUNCTION. SO WE’LL WRITE THIS
USING INEQUALITIES, AND WE’LL ALSO WRITE THIS USING
INTERVAL NOTATION. SO IF X IS LESS THEN OR=2
USING INTERVAL NOTATION, WE’D HAVE THE INTERVAL FROM
NEGATIVE INFINITY TO 2, CLOSE ON 2. SO THE SQUARE BRACKETS
INDICATE THAT 2 IS INCLUDED IN THIS INTERVAL. AND THEN FOR THE RANGE OR THE
POSSIBLE FUNCTION VALUES, FOR TAKING THE PRINCIPAL SQUARE
ROOT OF A NON NEGATIVE NUMBER, THEN THE FUNCTION VALUES
WOULD ALSO BE NON NEGATIVE. MEANING THAT Y WILL ALWAYS BE
GREATER THAN OR EQUAL TO ZERO, OR USING INTERVAL NOTATION, THE INTERVAL CLOSED ON ZERO
TO POSITIVE INFINITY. NOW, TO VERIFY
THE DOMAIN AND RANGE, LET’S GO AHEAD AND GRAPH
THE GIVEN FUNCTION. SO HERE’S A GRAPH OF F OF X. TO VERIFY THE DOMAIN
WE’LL PROJECT THIS GRAPH ON TO THE X AXIS. NOTICE THE LARGEST X VALUE 2, AND THEN WE CAN SEE X IS GOING
TO BE LESS THAN OR=2 ALONG THE X AXIS. SO THIS VERIFIES OUR DOMAIN. THEN TO VERIFY THE RANGE, IF WE
PROJECT THIS ON TO THE Y AXIS, NOTICE THE SMALLEST Y VALUE
IS 0. AND THEN THE GRAPH MOVES UP
FROM HERE, SO THE RANGE WILL BE
WHEN Y IS GREATER THAN OR=0. SO THIS GRAPH DOES VERIFY OUT
DOMAIN AND RANGE ARE CORRECT. NOW, LOOKING AT G OF X, BECAUSE WE HAVE THE PRINCIPAL
CUBE ROOT WHERE THE INDEX IS ODD, X – 1
CAN BE ANY REAL NUMBER. AND IF X – 1 CAN BE ANY REAL
NUMBER THEN SO CAN X. SO THE DOMAIN WOULD BE
ALL REAL NUMBERS OR USING INTERVAL NOTATION, WE’D HAVE THE INTERVAL
FROM NEGATIVE INFINITY TO POSITIVE INFINITY. SO IF WE TAKE THE CUBE ROOT OF POSITIVE AND NEGATIVE REAL
NUMBERS, AS WELL AS ZERO, THE FUNCTION VALUES FOR G OF X WOULD ALSO BE POSITIVE,
NEGATIVE, OR ZERO. AND THEREFORE, THE RANGE
IS ALSO ALL REAL NUMBERS. SO, AGAIN,
USING INTERVAL NOTATION, WE’D HAVE THE INTERVAL FROM NEGATIVE INFINITY
TO POSITIVE INFINITY. AGAIN, WE’LL GO AHEAD
AND GRAPH THIS FUNCTION TO VERIFY THE DOMAIN AND RANGE. HERE’S THE GRAPH OF G OF X. NOTICE IF WE PROJECT THIS
ON TO THE X AXIS THERE’S NO HOLES OR BRAKES, AND THE X VALUES WOULD BE FROM NEGATIVE INFINITY
TO POSITIVE INFINITY BECAUSE THE GRAPH MOVES RIGHT
AND LEFT INDEFINITELY. IF WE PROJECT THIS ON TO
THE Y AXIS TO VERIFY THE RANGE, EVEN THOUGH THE GRAPH
IS MOVING RIGHT VERY QUICKLY AND LEFT VERY QUICKLY, IT IS ALSO MOVING UPWARD
AND DOWNWARD WITHOUT ANY BRAKES. AND THEREFORE THE RANGE
WOULD ALSO BE ALL REAL NUMBERS. NOW FOR THE LAST FUNCTION, WE HAVE H OF X=THE FOURTH ROOT
OF 4X + 12. AND BECAUSE THE INDEX IS EVEN, OUR RADICAND 4X + 12
MUST BE GREATER THAN=0. SO WE’LL GO AHEAD
AND SOLVE FOR X. WE’LL SUBTRACT 12 ON BOTH SIDES. DIVIDE BY 4, SO WE HAVE X IS GREATER THAN
OR=-3, WHICH WOULD BE THE DOMAIN
OF OUR FUNCTION. USING INTERVAL NOTATION, WE’D HAVE THE INTERVAL CLOSE
ON -3 TO POSITIVE INFINITY. AND IF WE’RE TAKING THE FOURTH
ROOT OF A NON NEGATIVE NUMBER, THE FUNCTION VALUES WOULD ALSO
BE NON NEGATIVE. AND THEREFORE, THE RANGE IS
Y GREATER THAN=0, OR USING INTERVAL NOTATION, THE INTERVAL FROM 0 TO INFINITY,
CLOSE ON 0. AND, AGAIN, WE’LL VERIFY THIS
BY GRAPHING THE FUNCTION. HERE’S THE GRAPH OF H OF X. TO VERIFY THE DOMAIN, IF WE
PROJECT THIS ON TO THE X AXIS, NOTICE THE SMALLEST X VALUE
IS -3. AND FROM THERE IT MOVES TO THE
RIGHT, SO OUR DOMAIN WOULD BE X GREATER
THAN OR=-3. IF WE PROJECT THIS ON TO THE Y
AXIS, NOTICE HOW THE SMALLEST Y VALUE
IS 0, AND FROM HERE THE GRAPH
MOVES UPWARD, VERIFYING OUR RANGE IS Y
GREATER THAN OR=0. OKAY, THAT’S GOING TO DO IT
FOR THESE THREE EXAMPLES. I HOPE YOU FOUND
THIS EXPLANATION HELPFUL.  

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