- NOW WE'LL LOOK AT TWO EXAMPLES
OF DETERMINING IF FUNCTIONS ARE ODD, EVEN, OR NEITHER.
TO REVIEW, IF A FUNCTION IS EVEN,
THEN (F OF X) MUST EQUAL (F OF -X)
AND THE GRAPH WILL BE SYMMETRICAL ACROSS THE Y AXIS.
IF A FUNCTION IS ODD,
THEN (-F OF X) WILL EQUAL (F OF -X)
AND THE GRAPH WILL HAVE ROTATIONAL SYMMETRY
ABOUT THE ORIGIN,
WHICH MEANS IF WE ROTATE THE GRAPH
ABOUT THE ORIGIN 180 DEGREES, THE GRAPH WILL REMAIN UNCHANGED.
WHEN DETERMINING WHETHER A GRAPH IS ODD OR EVEN OR NEITHER
I ALWAYS LIKE TO LOOK AT THE GRAPH FIRST.
SO HERE'S THE GRAPH
OF (F OF X) = X TO THE FOURTH - (X SQUARED - 3).
RIGHT AWAY WE SHOULD BE ABLE TO RECOGNIZE
THAT IF WE WERE TO FOLD GRAPH ACROSS THE Y AXIS,
OR THIS LINE HERE,
THE TWO HALVES WOULD MATCH UP PERFECTLY
AND THEREFORE THIS HAS SYMMETRY ACROSS THE Y AXIS
AND THEREFORE THIS FUNCTION IS EVEN.
HOWEVER, WE'RE OFTEN ASKED TO SHOW THAT'S EVEN ALGEBRAICALLY,
WHICH MEANS YOU WANT TO SHOW THAT (F OF X) = -X.
SO LET'S GO AHEAD AND DO THAT.
LET'S DETERMINE (F OF -X)
AND SEE IF IT'S GOING TO BE THE SAME FUNCTION AS (F OF X).
SO (F OF -X) = -X RAISED TO THE FOURTH - (-X TO THE SECOND) - 3.
SO (F OF -X) IS EQUAL TO--
WELL, -X TO THE FOURTH IS GOING TO BE POSITIVE
BECAUSE WE HAVE AN EVEN NUMBER OF NEGATIVES.
SO THAT'LL BE X TO THE FOURTH
- (-X SQUARED) IS GOING TO BE (+X SQUARED - 3).
WE'LL NOTICE THAT (F OF -X)
AND (F OF X) ARE EQUAL TO EACH OTHER.
(F OF X) = (X TO THE FOURTH - X SQUARED - 3).
AND THEREFORE, ONCE AGAIN,
WE'VE SHOWN THAT THE GIVEN FUNCTION IS EVEN.
WE'VE SHOWN IT GRAPHICALLY.
WE'VE ALSO SHOWN IT ALGEBRAICALLY.
AND THERE IS ONE MORE THING TO MENTION,
IF WE TAKE A LOOK AT ALL THE VARIABLES ON THE EXPONENTS
AND IF WE HAD TO HAVE A VARIABLE ON 3 WE COULD WRITE THIS
AS 3X TO THE 0, SINCE X TO THE 0 = 1.
NOTICE ALL THE EXPONENTS ON ALL THE VARIABLES ARE EVEN,
0, 2, AND 4, AND WHEN THIS IS THE CASE,
THE POLYNOMIAL FUNCTION WILL ALWAYS BE EVEN,
WHICH WE'VE ALREADY SHOWN.
LET'S TAKE A LOOK AT A SECOND EXAMPLE.
HERE WE HAVE (F OF X) = (1/2X TO THE THIRD - 2X).
AGAIN, LET'S START BY LOOKING AT THE GRAPH OF THE FUNCTION,
WHICH WE CAN QUICKLY SEE
DOES NOT HAVE SYMMETRY ABOUT THE Y AXIS,
BUT IT DOES HAVE ROTATIONAL SYMMETRY ABOUT THE ORIGIN,
WHICH MEANS IF WE ROTATE THIS GRAPH ABOUT THE ORIGIN,
HALF A TURN, OR 180 DEGREES, IT WOULD REMAIN UNCHANGED.
THIS GREEN HALF WOULD BE WHERE THE BLUE HALF IS
AND THE BLUE HALF WOULD BE WHERE THE GREEN HALF IS.
SO BECAUSE OF THE SYMMETRY ABOUT THE ORIGIN
WE KNOW THIS FUNCTION IS GOING TO BE ODD,
BUT ALGEBRAICALLY WE WOULD NEED TO VERIFY THAT
(-F OF X) = (F OF -X) AND LET'S GO AHEAD AND DO THAT.
WE WANT TO SHOW THAT (-F OF X) = (F OF -X).
LET'S START WITH (F OF -X).
(F OF -X), WE'LL SUBSTITUTE -X FOR X IN OUR FUNCTION.
SO WE'D HAVE 1/2 -X RAISED TO THE THIRD - (2 x -X).
THIS WOULD BE -1/2X CUBED.
IF WE HAVE AN ODD NUMBER OF NEGATIVES
THE PRODUCT WILL BE NEGATIVE AND THIS IS GOING TO BE +2X.
WE WANT TO COMPARE THIS TO THE OPPOSITE OF (F OF X)
OR (-F OF X).
SO THE OPPOSITE OF (F OF X), WE'RE GOING TO CHANGE THE SIGN
OF EACH TERM ON THE RIGHT SIDE OF THIS FUNCTION.
SO WE WOULD HAVE -1/2X CUBED + 2X.
IF WE COMPARE THESE TWO FUNCTIONS,
NOTICE HOW THEY'RE EQUAL,
THEREFORE WE HAVE JUST VERIFIED THAT (-F OF X) = (F OF -X),
THEREFORE THE FUNCTION IS ODD,
WHICH WE ALREADY KNEW GRAPHICALLY.
AND THERE'S ONE MORE THING TO MENTION HERE,
BECAUSE WE HAVE A POLYNOMIAL FUNCTION
AND EACH TERM HAS AN ODD EXPONENT ON THE VARIABLE,
THIS WOULD BE X TO THE FIRST.
THIS INDICATES THE POLYNOMIAL FUNCTION WOULD BE ODD.
WE'LL TAKE A LOOK AT TWO MORE EXAMPLES IN THE NEXT VIDEO.