- WELCOME TO AN INTRODUCTORY LESSON
ON EVEN AND ODD FUNCTIONS.
THERE ARE TWO MAIN WAYS TO DETERMINE
IF A FUNCTION IS EVEN OR ODD.
ONE IS AN ALGEBRAIC METHOD, AND THE OTHER IS A GRAPHICAL METHOD.
A FUNCTION IS EVEN IF ALGEBRAICALLY F OF X = F OF -X,
AND GRAPHICALLY THE GRAPH HAS SYMMETRY ACROSS THE Y AXIS.
AND IF WE'RE WORKING WITH A POLYNOMIAL FUNCTION,
A POLYNOMIAL FUNCTION WILL HAVE ALL EVEN EXPONENTS
ON THE VARIABLES.
FOR ODD FUNCTIONS ALGEBRAICALLY, -F OF X = F OF -X,
AND GRAPHICALLY THE GRAPH HAS ROTATIONAL SYMMETRY
ABOUT THE ORIGIN,
WHICH MEANS THE GRAPH REMAINS THE SAME
AFTER A ROTATION OF 180 .
A POLYNOMIAL FUNCTION THAT IS ODD
WILL HAVE ALL ODD EXPONENTS ON THE VARIABLE.
LET'S TAKE A LOOK AT TWO BASIC EXAMPLES
TO GET AN IDEA OF WHAT'S HAPPENING HERE.
WE'LL CONSIDER F OF X = X SQUARED AND F OF X = X CUBED.
LOOKING AT THE GRAPH OF F OF X = X SQUARED,
NOTICE THAT IF WE WERE TO FOLD THIS GRAPH ACROSS THE Y AXIS
WE WOULD MATCH UP PERFECTLY WITH THE OTHER HALF.
WHICH MEANS THIS GRAPH HAS SYMMETRY ACROSS THE Y AXIS,
AND THEREFORE F OF X = X SQUARED IS AN EVEN FUNCTION.
IF WE WANT TO SHOW THAT IT'S EVEN ALGEBRAICALLY
WE WOULD HAVE TO SHOW THAT F OF X = F OF -X.
LET'S GO AHEAD AND DO THAT.
AGAIN, TO SHOW THIS FUNCTION IS EVEN ALGEBRAICALLY
WE WANT TO SHOW THAT F OF X = F OF -X.
WELL WE'RE GIVEN F OF X, SO LET'S DETERMINE F OF -X.
TO DETERMINE F OF -X
THE INPUT INTO OUR FUNCTION IS GOING TO BE -X
INSTEAD OF JUST X.
SO WE'D HAVE -X SQUARED.
WELL A NEGATIVE x A NEGATIVE IS POSITIVE,
SO THIS IS EQUAL TO X SQUARED.
SO F OF -X = X SQUARED, AND WE ALREADY KNOW F OF X = X SQUARED.
SO WE'VE JUST SHOWN ALGEBRAICALLY
THAT THE FUNCTION IS EVEN.
SO SINCE F OF X = F OF -X, THE FUNCTION IS EVEN.
ONE MORE THING WE COULD MENTION ABOUT THIS BASIC FUNCTION
IS IT'S A POLYNOMIAL FUNCTION, AND ALL THE EXPONENTS ARE EVEN.
THEREFORE IT'S ALSO AN EVEN FUNCTION.
NOW LET'S CONSIDER F OF X = X CUBED.
TO DETERMINE GRAPHICALLY IF THIS IS AN ODD FUNCTION,
THE GRAPH WOULD HAVE TO HAVE
ROTATIONAL SYMMETRY ABOUT THE ORIGIN.
WHICH MEANS IF WE ROTATE THIS 180 , OR HALF A TURN,
ABOUT THE ORIGIN, THE GRAPH SHOULD REMAIN UNCHANGED.
SO IF WE ROTATE THIS 180 ,
NOTICE HOW THIS PIECE HERE IN THE FIRST QUADRANT
WOULD MATCH UP PERFECTLY WITH THIS PIECE HERE
IN THE THIRD QUADRANT.
AND THE PIECE IN THE THIRD QUADRANT
WOULD END UP IN THE FIRST QUADRANT,
AND THE GRAPH WOULD LOOK EXACTLY THE SAME.
THEREFORE F OF X = X CUBED IS AN ODD FUNCTION GRAPHICALLY.
BUT TO VERIFY THIS ALGEBRAICALLY
WE NEED TO SHOW THAT -F OF X = F OF -X,
AND LET'S GO AHEAD AND DO THAT AS WELL.
SO WE WANT TO SHOW -F OF X = F OF -X.
LET'S START WITH THE RIGHT SIDE, AND DETERMINE F OF -X.
SO WE'RE GOING TO REPLACE X WITH -X IN OUR FUNCTION,
SO WE'D HAVE -X TO THE 3rd.
WELL, HERE WE'D HAVE 3 NEGATIVES BEING MULTIPLIED TOGETHER,
SO THIS ENDS UP BEING -X CUBED.
AND THEN FOR -F OF X,
THIS MEANS WE'RE GOING TO CHANGE THE SIGN OF OUR FUNCTION.
SO IF WE'RE TAKING THE OPPOSITE OF F OF X,
WE WOULD TAKE THE OPPOSITE OF THE RIGHT SIDE AS WELL,
SO THIS WOULD EQUAL -X CUBED.
AND NOTICE THAT -F OF X = F OF -X.
THEY'RE BOTH -X CUBED.
THEREFORE WE HAVE VERIFIED THAT THIS FUNCTION IS ODD.
WE'RE GOING TO GO AHEAD AND STOP HERE ON THIS INTRODUCTORY VIDEO.
WE'LL TAKE A LOOK AT SEVERAL MORE EXAMPLES OF DETERMINING
IF A GIVEN FUNCTION IS EVEN OR ODD IN THE NEXT FEW VIDEOS.
SO IF YOU WANT TO,
YOU CAN FOLLOW THE LINKS TO THE ADDITIONAL EXAMPLES
ON THE SCREEN.
I HOPE YOU FOUND THIS HELPFUL.