- WELCOME TO PART TWO OF AN
"INTRODUCTION TO FUNCTIONS." IN THIS VIDEO, WE'LL TAKE A LOOK
AT GRAPHING A FUNCTION AND ALSO HOW TO IDENTIFY
THE DOMAIN AND THE RANGE OF A FUNCTION. THE GOOD NEWS IS THAT
GRAPHING A FUNCTION IS JUST LIKE GRAPHING
ANY OTHER EQUATION THAT WE'VE DONE IN THE PAST, MEANING IF WE HAVE THE FUNCTION
F OF X = 4X - 2, IF WE WANT TO, WE COULD REWRITE
THIS AS Y = 4X - 2. REMEMBER, F OF X = Y, SO IF WE
WANT TO GRAPH THIS FUNCTION, WE HAVE A VARIETY OF OPTIONS. THIS IS IN SLOPE-INTERCEPT FORM, OR WE CAN GO BACK TO MAKING
A T-TABLE. JUST FOR ILLUSTRATION, LET'S GO
AHEAD AND MAKE A T-TABLE. REMEMBER, WE CAN PICK
ANY X VALUE THAT WE WANT. LET'S GO AHEAD AND SELECT
0, 1, 2, AND THEN MAYBE -1. SO NOW, WE'RE JUST GOING
TO REPLACE X WITH THESE VALUES IN THIS EQUATION TO DETERMINE Y. SO 4 x 0 - 2 = -2,
4 x 1 - 2 = 2. AND WHEN X IS 2,
WE'D HAVE 4 x 2 = 8 - 2 = 6. AND THEN WHEN X = -1,
WE HAVE -4 - 2 = -6. LET'S GO AHEAD
AND PLOT THESE POINTS. WE HAVE (0,-2), (1,2), (2,6),
AND (-1,-6). SO FOR THE MOST PART, GRAPHING THIS FUNCTION
SHOULD BE REVIEW, BECAUSE WE'VE BEEN GRAPHING
LINES NOW FOR QUITE SOME TIME. HOWEVER, I DO WANT TO MAKE
THE CONNECTION THAT IF WE HAVE THE (.0,-2), IF WE WANT TO REWRITE THIS
USING FUNCTION NOTATION, WE WOULD SAY THAT F OF 0 = -2. 0 IS THE INPUT OF THE X VALUE, AND THE ENTIRE FUNCTION VALUE
OR Y = -2. FOR THE .12, THAT IMPLIES
THAT F OF 1 = +2. SO WE SAY F OF 1 = 2 WHICH MEANS
WHEN THE INPUT X = 1, THE OUTPUT OR THE Y VALUE = 2. AND FOR THE (.2,6),
WE WOULD SAY THAT F OF 2 = 6, MEANING WHEN THE INPUT IS 2,
THE OUTPUT IS 6. OKAY. AND THIS PROBLEM ALSO ASKS US
TO DETERMINE THE DOMAIN AND THE RANGE. REMEMBER, THE DOMAIN IS A SET OF
ALL POSSIBLE INPUTS OR X VALUES. AND WE KNOW WE CAN SELECT ANY
REAL NUMBER FOR THE VALUE OF X, SO THE DOMAIN WOULD BE
ALL REAL NUMBERS. LET'S TAKE A LOOK AT IT
GRAPHICALLY. X VALUES ARE THE VALUES
THAT MOVE FROM LEFT TO RIGHT. SO AS WE CAN SEE,
THIS LINE IS SLANTED, BUT IT DOES MOVE LEFT AND RIGHT
FOREVER VERIFYING THAT OUR DOMAIN
IS ALL REALS. NOW, THE RANGE IS THE SET
OF POSSIBLE OUTPUTS OR Y VALUES, AND GRAPHICALLY
THIS IS THE Y-AXIS, SO Y VALUES MOVE UP AND DOWN. AND WE CAN SEE GRAPHICALLY THAT
THIS LINE DOES GO DOWN FOREVER AND ALSO GOES UP FOREVER, AND THAT THERE ARE INDICATIONS THAT THE RANGE
IS ALSO ALL REALS. LET'S TAKE A LOOK
AT ANOTHER ONE. HERE WE HAVE F OF X = X SQUARED, AND AGAIN WHEN WE FIRST START
TO WORK WITH FUNCTIONS, IT MAY BE HELPFUL TO REWRITE
THIS AS Y = X SQUARED. SO LET'S GO AHEAD
AND MAKE A T-TABLE, AND LET'S SELECT
-2, -1, 0, 1, 2. WELL, ONCE WE HAVE OUR X VALUE,
TO FIND Y, WE HAVE TO SQUARE X. SO -2 SQUARED = 4,
-1 SQUARED = 1, 0 SQUARED = 0, 1 SQUARED = 1,
AND 2 SQUARED = 4. LET'S GO AHEAD
AND PLOT THESE POINTS AND SEE WHAT KIND OF GRAPH
WE HAVE. (-2,4) IS HERE, (-1,1) IS HERE,
(0,0), (1,1), AND (2,4). HERE WE HAVE A U-SHAPED GRAPH
WHICH IS CALLED A PARABOLA, AND LET'S SEE IF WE CAN
DETERMINED THE DOMAIN AND THE RANGE OF THIS FUNCTION. WELL, AGAIN THE DOMAIN IS
THE SET OF INPUTS OR X VALUES. WE CAN PICK ANY REAL NUMBER
AND SQUARE IT, SO AGAIN THE DOMAIN
WOULD BE ALL REALS. GRAPHICALLY,
EVEN THOUGH IT'S PRETTY CLEAR THIS GRAPH MOVES UP FOREVER, IT IS ALSO SLOWLY CREEPING LEFT
AND CREEPING RIGHT WHICH DOES VERIFY THAT
OUR DOMAIN IS ALL REALS. HOWEVER IF WE LOOK AT OUR RANGE,
SINCE WE'RE SQUARING X, EVEN WHEN X IS NEGATIVE,
Y IS ALWAYS POSITIVE OR 0. SO Y IS ALWAYS GOING TO BE
GREATER THAN OR EQUAL TO 0, AND THAT IS GOING TO BE
OUR RANGE. GRAPHICALLY, WE CAN SEE THAT
HERE'S THE Y-AXIS, AND THE LOWEST POINT ON THE
GRAPH IS RIGHT HERE WHEN Y = 0. ALL THE OTHER Y VALUES
ARE POSITIVE VERIFYING THAT OUR RANGE WOULD BE Y
GREATER THAN OR EQUAL TO 0. WHENEVER WE'RE GRAPHING
A FUNCTION, WE CAN ALWAYS VERIFY IT
IN THE GRAPHING CALCULATOR, BECAUSE REMEMBER F OF X = Y. SO WHEN WE HAVE FUNCTION
NOTATION, THE EQUATION IS ALREADY
SOLVED FOR Y. LET'S GO AHEAD AND VERIFY THIS
ONE IN THE GRAPHING CALCULATOR. SO WE'LL PRESS "Y=." WE'RE JUST GOING TO TYPE
IN THE RIGHT SIDE FOR Y1. WE'LL TYPE IN "X SQUARED."
SO HERE'S X. HERE'S THE SQUARED KEY. WE'LL GO AHEAD AND PRESS
"GRAPH." THERE'S THE GRAPH
OF OUR PARABOLA THAT WE HAVE HERE IN RED. IF WE WANT TO FIND SPECIFIC
POINTS TO VERIFY OUR T-TABLE, WE CAN PRESS "2nd GRAPH," AND WE CAN VERIFY THAT WE HAVE
COMPLETED OUR T-TABLE CORRECTLY FROM THE SCREEN HERE. LET'S GO AHEAD
AND TAKE A LOOK AT ONE MORE. HERE WE HAVE F OF X =
THE ABSOLUTE VALUE OF X - 3. AND AGAIN IF WE WANT,
WE CAN REWRITE THIS AS Y = THE ABSOLUTE VALUE OF X - 3. LET'S GO AHEAD AND COMPLETE
A T-TABLE FOR THIS. LET'S GO AHEAD AND USE
-2, -1, 0, 1, 2. OKAY. SO WHEN X = -2, THE ABSOLUTE
VALUE OF -2 = +2 - 3 = -1. THE ABSOLUTE VALUE OF -1 = 1
-3 = -2. THE ABSOLUTE VALUE OF 0
= 0 - 3 = -3. THE ABSOLUTE VALUE OF 1
= 1 - 3 = -2, AND THE ABSOLUTE VALUE
OF 2 = 2 -3 = -1. LET'S GO AHEAD
AND PLOT THESE POINTS. WE HAVE (-2,-1), (-1,-2),
(0,-3), (1,-2), AND (2,-1). SO DEPENDING ON YOUR EXPERIENCE GRAPHING ABSOLUTE VALUE
EQUATIONS, THIS PRODUCES A V-SHAPED GRAPH, AS WE SEE HERE. LET'S SEE IF WE CAN FIGURE OUT WHAT THE DOMAIN AND RANGE
WOULD BE. AGAIN, REMEMBER THE DOMAIN
THE SET OF ALL X VALUES FOR THIS FUNCTION. AND AGAIN, WE CAN SELECT ANY
REAL NUMBER FOR THE VALUE OF X, SO THE DOMAIN
WOULD BE ALL REALS. GRAPHICALLY, REMEMBER THE X-AXIS
IS THE HORIZONTAL AXIS MOVING LEFT AND RIGHT, AND THIS FUNCTION DOES MOVE
RIGHT FOREVER AND LEFT FOREVER INDICATING THE DOMAIN
IS ALL REALS. NOW, THE RANGE IS A SET
OF ALL Y VALUES. REMEMBER THE Y-AXIS
IS THE VERTICAL AXIS, SO WE WANT TO ASK OURSELVES, "HOW DOES THIS FUNCTION BEHAVE
MOVING UP AND DOWN?" WELL, IT DOES GO UP FOREVER,
BUT IT DOESN'T GO DOWN FOREVER. IN FACT, THE LOWEST Y VALUE
IS -3, SO OUR RANGE IS GOING TO BE
Y GREATER THAN OR EQUAL TO -3. LET'S GO AHEAD AND SHOW
HOW GRAPH THIS ON THE CALCULATOR, AS WELL. LET'S GO BACK TO "Y=."
LET'S CLEAR THIS OUT. TO ACCESS THE ABSOLUTE VALUE
FEATURE, WE PRESS "MATH," RIGHT ARROW ONCE TO "NUMBER," AND THEN "ABS" STANDS FOR
ABSOLUTE VALUE, SO WE'LL PRESS "ENTER." IT BRINGS US BACK
TO THIS SCREEN. WE PRESS "X," CLOSE PARENTHESIS,
AND THEN - 3, GRAPH. THIS VERIFIES OUR GRAPH. AND THEN TO FIND POINTS
ON THE GRAPH, WE CAN PRESS "2nd GRAPH"
FOR THE T-TABLE. BEFORE WE FINISH, LET'S GO AHEAD AND GO BACK
TO THESE ORDERED PAIRS AND REPRESENT THEM
USING FUNCTION NOTATION. WHEN X IS -2, Y = -1. SO USING FUNCTION NOTATION,
WE WOULD SAY, F OF -2 = -1. WHEN THE INPUT INTO THE FUNCTION
IS -2, THE OUTPUT IS -1. THE INPUT IS X,
AND THE OUTPUT IS Y. SO FOR THE (.-1,-2),
WE HAVE F OF -1. THERE'S OUR INPUT, AND THEN THE OUTPUT WILL BE
THE Y VALUE OR -2. HERE WE'D HAVE F OF 0 = -3,
HERE WE'D HAVE F OF 1 = -2, AND SO ON. OKAY, THAT'S GOING TO DO IT
FOR PART TWO. THERE IS ANOTHER VIDEO ON THE
DOMAIN AND RANGE OF FUNCTIONS. I DO REALIZE THAT WHEN WE FIRST
INTRODUCED DOMAIN AND RANGE, IT CAN BE A LITTLE BIT
CHALLENGING. I HOPE YOU FOUND THIS VIDEO
HELPFUL.