The Algebra of Functions

The Algebra of Functions


– TODAY WE WILL BE TALKING ABOUT
THE ALGEBRA OF FUNCTIONS. THE GOAL IS TO FIND THE SUM,
DIFFERENCE, PRODUCT, AND QUOTIENT OF TWO FUNCTIONS. IF WE LET F AND G BE FUNCTIONS
AND X IS IN THE DOMAIN OF BOTH, F + G OF X=F OF X + G OF X. F – G OF X=F OF X – G OF X. F x G OF X=F OF X x G OF X. AND F DIVIDED BY G OF X=F OF X
DIVIDED BY G OF X. SO LETS TAKE A LOOK AT HOW
WE ARE GOING TO USE THIS. IF WE’RE GIVEN TWO FUNCTIONS WE
SHOULD BE ABLE TO FIND THE SUM, DIFFERENCE, PRODUCT, AND
QUOTIENT OF THE FUNCTIONS AT A SPECIFIC VALUE OF X. SO BY APPLYING THESE
DEFINITIONS, WE KNOW THAT F + G OF -2=F
OF -2 + G OF -2. SO NOW WE JUST HAVE TO PERFORM
OUR SUBSTITUTION INTO EACH FUNCTION
AND THEN FIND THE SUM. TO FIND F OF -2 WE HAVE TO REPLACE X WITH -2
AND FUNCTION F. THAT WOULD LOOK LIKE THIS + G OF
-2, WHICH WOULD LOOK LIKE THIS. SIMPLIFYING EACH OF THESE,
HERE WE’D HAVE 8 + 3=11. AND HERE WE’D HAVE -2 SQUARED,
WHICH IS 4 – 3=1. SO F + G OF -2=12. THE POINT IS NOT REALLY TO DO A
BUNCH OF NUMBER CRUNCHING, SO LET’S GO AHEAD AND USE THE
GRAPHING CALCULATOR TO ASSIST US ON THESE NEXT THREE. SO WHAT WE CAN DO IS TYPE F INTO
Y1 AND G INTO Y2. NOW LET’S MAKE SURE OUR TABLE
IS SET ON ASK, SO LETS PRESS SECOND WINDOW, SCROLL DOWN TO INDEPENDENT
VARIABLE, HIGHLIGHT ASK, AND HIT ENTER. NOW LETS GO TO THE T TABLE,
SECOND GRAPH. SO FOR EXAMPLE,
ON THIS FIRST PROBLEM, IF WE LOOK AT THE VALUES OF THE
FUNCTIONS AT X=-2, IF WE PRESS -2 WE CAN SEE THAT
Y1 + Y2=12, WHICH IS WHAT WE FOUND FOR THE
ANSWER FOR THIS FIRST PROBLEM. NOW, GOING BACK TO OUR EXAMPLES,
F – G OF 5 IS EQUAL TO– HERE ARE OUR RULES,
F OF 5 – G OF 5. WELL, LET’S GO BACK TO OUR
GRAPHING CALCULATOR AND FIND THESE VALUES. SO IF WE PRESS 5 ENTER, REMEMBER
THAT Y1 IS F AND Y 2 IS G. SO HERE WE’D HAVE -17 – 22,
WHICH=-39. THAT WOULD BE F -G OF 5. ON THIS NEXT PROBLEM, F x G OF 0
=F OF 0 x G OF 0. WELL, THIS ONE WE CAN PROBABLY DO WITHOUT THE GRAPHING
CALCULATOR. F OF 0=3, G OF 0,
0 SQUARED – 3=-3. WELL, 3 x -3, OF COURSE, IS -9. F x G OF 0=-9
AND THE LAST PROBLEM, F DIVIDED BY G OF 2=F OF 2
DIVIDED BY G OF 2. SO IF WE GO TO THE GRAPHING
CALCULATOR, PRESS 2 FOR X, WE WOULD HAVE F OF 2=-5,
G OF 2=1. SO -5/1, OF COURSE, IS=-5. F DIVIDED BY G OF 2=-5. NOW, ANOTHER WAY TO APPROACH
THIS SAME IDEA IS BY LOOKING AT IT GRAPHICALLY. SO IF WE FIRST APPLY
THE DEFINITION, F + G OF 8=F OF 8 + G OF 8. REMEMBER THESE ARE FUNCTION
VALUES; THEREFORE WE’RE TALKING
ABOUT Y VALUES. F OF 8, HERE’S OUR FUNCTION F. HERE’S X=8, F OF 8
=POSITIVE 3, G OF 8=-4, SO OUR SUM=-1. ON THE NEXT PROBLEM, F x G OF -4
=F OF -4 x G OF -4. LET’S FIND THESE FUNCTION VALUES
FROM THE GRAPH. F OF -4,
HERE’S F, HERE’S F=-4. THE Y VALUE, OR THE FUNCTION
VALUE,=0 x G OF -4. G IS IN RED, G OF -4=2,
WHICH, OF COURSE, WOULD GIVE US 0. AND ON THIS
LAST EXAMPLE, WE DO NOT HAVE ENOUGH
INFORMATION TO ANSWER THIS QUESTION BECAUSE WE DO NOT HAVE THE
EQUATIONS FOR THE FUNCTIONS. THANK YOU FOR WATCHING
AND HAVE A GOOD DAY.  

6 thoughts on “The Algebra of Functions

  • November 30, 2011 at 9:51 pm
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    Thank you!

    Reply
  • September 28, 2013 at 1:43 am
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    Your'e Awesome !!! đŸ˜€

    Reply
  • October 9, 2014 at 12:55 pm
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    no me gusta la clase de matemĂ¡ticas

    Reply
  • October 9, 2014 at 12:57 pm
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    marco es muy guapo

    Reply
  • October 9, 2014 at 12:57 pm
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    brett es muy bonita

    Reply
  • September 26, 2019 at 7:47 pm
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    atleast show your work

    Reply

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