- WELCOME TO A LESSON ON FUNCTION NOTATION.
NOW THAT WE KNOW A FUNCTION IS A SPECIAL RELATION
IN WHICH EACH INPUT HAS EXACTLY ONE OUTPUT;
WE OFTEN USE SPECIAL NOTATION TO COMMUNICATE THAT A RELATION
IS A FUNCTION.
THE NOTATION IS CALLED FUNCTION NOTATION.
FOR EXAMPLE, HERE WE SEE Y = F OF X.
SO F OF X IS FUNCTION NOTATION WHERE F OF X IS THE OUTPUT
RESULTING FROM APPLYING THE FUNCTION RULE
F TO THE INPUT X.
SO WHEN USING FUNCTION NOTATION WE NEED TO REMEMBER
THAT X IS THE INPUT AND F OF X IS THE OUTPUT
AND THEREFORE F OF X IS = TO Y.
WHEN WE FIRST SEE THIS NOTATION THOUGH IT'S EASY TO THINK
THAT THIS MEANS F x X, BUT IT DOES NOT.
IT'S SPECIAL NOTATION CALLED FUNCTION NOTATION.
AS AN EXAMPLE, IF WE WERE GIVEN THE FUNCTION F OF X = 2X + 3
THE FIRST THING WE SHOULD RECOGNIZE IS THIS IS EQUIVALENT
TO THE EQUATION Y = 2X + 3.
BUT WE USE THIS SPECIAL FUNCTION NOTATION TO INDICATE
THAT THIS EQUATION DOES REPRESENT A FUNCTION.
TO FIND A FUNCTION VALUE, FOR EXAMPLE, LETS SAY F OF 4,
WE SUBSTITUTE 4 FOR X.
SO F OF 4 IS = TO 2 x 4 INSTEAD OF 2 x X AND THEN + 3.
WELL, 2 x 4 IS 8 SO THIS IS = TO 8 + 3
WHICH IS = TO 11.
SO WE SAY F OF 4 = 11, BUT WE SHOULD ALSO RECOGNIZE
THIS REPRESENTS A POINT ON THE GRAPH OF OUR FUNCTION
WHERE THE X COORDINATE IS 4 AND THE Y COORDINATE IS 11.
SO WHEN WE FIND FUNCTION VALUES, NOTICE HOW WE'RE FINDING
Y VALUES.
LET'S TAKE A LOOK AT SEVERAL MORE EXAMPLES.
HERE WE ARE GIVEN F OF X AND ASKED TO FIND F OF 4
AND F OF -3.
SO THIS REALLY COMES DOWN TO SUBSTITUTING 4 FOR X
AND THEN -3 FOR X.
SO F OF 4 WOULD BE = TO INSTEAD OF X SQUARED
WE'D HAVE 4 SQUARED, INSTEAD OF - 5 x X
WE'LL HAVE - 5 x 4 AND THEN + 6.
AND NOW WE SIMPLIFY THIS FOLLOWING
THE ORDER OF OPERATIONS.
SO WE'D FIRST SQUARE 4, THAT'S 16 AND THEN - 5 x 4
THAT'D BE - 20 + 6.
SO THIS WOULD BE 16 - 20 IS -4 + 6 IS +2.
SO WE CAN SAY F OF 4 = +2, WHICH AGAIN IT WOULD REPRESENT
THE POINT 4, 2 ON THE GRAPH
OF THIS GIVEN FUNCTION.
FOR F OF -3 WE'LL NOW SUBSTITUTE
-3 FOR X SO WE'D HAVE -3 SQUARED - 5 x -3 + 6.
IT IS IMPORTANT THAT WE INCLUDE PARENTHESIS AROUND
THIS -3, X SQUARED MEANS X x X SO FOR -3 SQUARED WE DO NEED
-3 x -3, WHICH WOULD BE +9.
SO WE WOULD HAVE +9.
HERE WE HAVE -5 x -3 THAT'S + 15 AND THEN + 6
SO THIS WOULD BE 9 + 15 THAT'S 24 + 6 IS 30.
SO WE WOULD SAY F OF -3 = 30.
LET'S TAKE A LOOK AT 2 MORE EXAMPLES.
NOTICE, IN THIS EXAMPLE OUR FUNCTION IS G OF T.
SO THE INPUT IS T AND THE OUTPUT IS G OF T.
SO TO FIND G OF -3 WE'LL SUBSTITUTE -3 FOR T,
BUT WE NEED TO BE CAREFUL WHEN PERFORMING
THIS SUBSTITUTION.
WE CAN THINK OF THIS AS THE OPPOSITE OF T SQUARED,
SO IF WE SUBSTITUTE -3 FOR T WE WOULD HAVE NEGATIVE
AND THEN -3 SQUARED + 2 x -3 - 12.
SO WHEN WE BEGIN TO SIMPLIFY HERE -3 SQUARED IS +9,
BUT THEN WE HAVE A NEGATIVE SIGN HERE SO THAT IS GOING TO BE
-9 AND WE HAVE + 2 x -3 THAT'S + -6 OR JUST - 6 - 12.
THIS WOULD BE -15 - 12 THAT'S -27 SO G OF -3 = -27.
SO WE HAVE AN INPUT OF T = -3, OUR FUNCTION VALUE
OR OUTPUT IS -27.
NOTICE IN OUR NEXT EXAMPLE THE INPUT IS A VARIABLE
EXPRESSION 3X, WHICH MEANS G OF 3X
WILL NOT GIVE US A REAL VALUE.
IT WILL GIVE US ANOTHER FUNCTION,
BUT THE PROCESS IS THE SAME.
WE'RE GOING TO SUBSTITUTE 3X FOR T.
SO WE'LL HAVE NEGATIVE AND THEN 3X SQUARED + 2 x 3X - 12.
WELL, 3X SQUARED WOULD BE 9X SQUARED.
THEN WE WANT THE NEGATIVE OF THAT OR THE OPPOSITE OF THAT.
SO -9X SQUARED, THIS WILL BE + 6X AND THEN - 12.
WE CAN'T COMBINE ANY OF THESE TERMS
BECAUSE THEY'RE NOT LIKE TERMS SO WE'RE ACTUALLY DONE.
G OF 3 X IS = TO -9X SQUARED + 6X - 12.
LET'S GO AHEAD AND TAKE A LOOK AT ONE MORE EXAMPLE.
HERE WE ARE GIVEN THE FUNCTION H OF Y = 2 Y SQUARED - Y + 7
AND WE'RE ASKED TO FIND H OF THE QUANTITY M - 2.
SO IN THIS CASE WE'LL SUBSTITUTE THE QUANTITY M - 2 FOR Y.
SO THIS WOULD GIVE US 2 x THE QUANTITY M - 2 SQUARED
- THE QUANTITY M - 2 + 7.
FIRST, WE NEED TO SIMPLIFY THE EXPONENTS.
SO WE'LL HAVE TO MULTIPLY M - 2 BY ITSELF.
THAT WOULD GIVE US 2 x M - 2 x M - 2.
LET'S GO AHEAD AND MULTIPLY THIS OUT FIRST BY 4 PRODUCT'S
1, 2, 3, 4.
SO WE'LL HAVE 2 x THE QUANTITY M x M IS M SQUARED
AND THEN WE HAVE M x -2 THAT'S -2M.
AND THEN WE HAVE -2 x M, THAT'S ANOTHER -2 M
SO - 4M -2 x -2 IS + 4 AND NOW WE'LL CLEAR THE PARENTHESIS.
HERE, WE'LL DISTRIBUTE 2 AND BECAUSE OF THE MINUS SIGN
HERE, WE CAN THINK OF DISTRIBUTING A -1.
SO WE WOULD HAVE 2M SQUARED - 8M + 8 AND THEN WE'D HAVE
-M + 2 AND + 7.
FOR THE LAST STEP WE'LL COMBINE LIKE TERMS.
THERE'S ONLY 1 M SQUARED TERM, THERE ARE 2 M TERMS,
-8M - 1M IS -9M OR - 9M AND THEN WE HAVE 8 + 2 + 7,
WHICH IS 17.
SO WE FOUND THAT H OF M - 2 IS = TO 2M SQUARED - 9M + 17.
I HOPE YOU FOUND THIS HELPFUL.