Operation of functions and Composite function.pdf

Prepared by:
Mr. George G. Lescano

Addition of Function:
Tip 1: Combining like terms.
Tip 3: Be careful with the integers.
Tip 2: Put the equation/s in DESCENDING
ORDERS.

Addition of Function:
&#3627408519;&#3627408537;=&#3627408537;
&#3627409360;
−&#3627409361;&#3627408514;&#3627408527;&#3627408517;&#3627408520;&#3627408537;=&#3627409360;&#3627408537;+&#3627409363;Example 1: −&#3627409361;+5
&#3627408537;
&#3627409360;
−3
&#3627409360;&#3627408537;+&#3627409363;
+
+&#3627409360;&#3627408537;
&#3627409360;
+2x
&#3627408537;
&#3627409360;
+&#3627409360;&#3627408537;−&#3627409361;+&#3627409363;
&#3627408537;
&#3627409360;
+&#3627409360;&#3627408537;+&#3627409360;
&#3627408467;+&#3627408468;??????=??????
2
+2??????+2

Addition of Function:
Example 2:
&#3627408514;&#3627408537;=&#3627408537;
&#3627409360;
−&#3627409361;&#3627408537;+&#3627409359;−&#3627409360;&#3627408537;
&#3627409361;
&#3627408514;&#3627408527;&#3627408517;&#3627408515;&#3627408537;=−&#3627409363;&#3627408537;+&#3627408537;
&#3627409361;
−&#3627409359;&#3627409358;
&#3627408514;&#3627408537;=−&#3627409360;&#3627408537;
&#3627409361;
+&#3627408537;
&#3627409360;
−&#3627409361;&#3627408537;+&#3627409359;&#3627408515;&#3627408537;=&#3627408537;
&#3627409361;
−&#3627409363;&#3627408537;−&#3627409359;&#3627409358;
−&#3627409360;&#3627408537;
&#3627409361;
+&#3627408537;
&#3627409360;
−&#3627409361;&#3627408537;+&#3627409359;
&#3627408537;
&#3627409361;+
−&#3627409363;&#3627408537;−&#3627409359;&#3627409358;
−&#3627408537;
&#3627409361;
+&#3627408537;
&#3627409360;
−&#3627409366;&#3627408537;−&#3627409367;
&#3627408514;+&#3627408515;&#3627408537;=−&#3627408537;
&#3627409361;
+&#3627408537;
&#3627409360;
−&#3627409366;&#3627408537;−&#3627409367;

Addition of Function:
Using the same given in example 2, find:
&#3627408514;&#3627408537;=&#3627408537;
&#3627409360;
−&#3627409361;&#3627408537;+&#3627409359;−&#3627409360;&#3627408537;
&#3627409361;
&#3627408514;&#3627408527;&#3627408517;&#3627408515;&#3627408537;=−&#3627409363;&#3627408537;+&#3627408537;
&#3627409361;
−&#3627409359;&#3627409358;
(&#3627408514;+&#3627408515;)(−&#3627409361;)
&#3627408514;+&#3627408515;−&#3627409361;=&#3627409363;&#3627409359;
&#3627408514;+&#3627408515;&#3627408537;=−&#3627408537;
&#3627409361;
+&#3627408537;
&#3627409360;
−&#3627409366;&#3627408537;−&#3627409367;
=−−&#3627409361;
&#3627409361;
+−&#3627409361;
&#3627409360;
−&#3627409366;(−&#3627409361;)−&#3627409367;
=−−&#3627409360;&#3627409365;+&#3627409367;+&#3627409360;&#3627409362;−&#3627409367;
=&#3627409360;&#3627409365;+&#3627409367;+&#3627409360;&#3627409362;−&#3627409367;

Subtraction of Function:
fgxf xgx
CAUTION:Makesureyoudistributethe–toeach
termofthesecondfunction.Youshouldsimplify
bycombiningliketerms.

Subtraction of Function:
Example 1:&#3627408519;&#3627408537;=&#3627409361;&#3627408537;
&#3627409360;
+&#3627409359;&#3627409358;&#3627408514;&#3627408527;&#3627408517;&#3627408520;&#3627408537;=&#3627409362;&#3627408537;+&#3627409365;
&#3627409361;&#3627408537;
&#3627409360;
+&#3627409359;&#3627409358;(&#3627409362;&#3627408537;+&#3627409365;)−
&#3627409361;&#3627408537;
&#3627409360;
+&#3627409359;&#3627409358;−&#3627409362;&#3627408537;−&#3627409365;
&#3627409361;&#3627408537;
&#3627409360;
−&#3627409362;&#3627408537;+&#3627409359;&#3627409358;−&#3627409365;
&#3627409361;&#3627408537;
&#3627409360;
−&#3627409362;&#3627408537;+&#3627409361;
&#3627408519;−&#3627408520;&#3627408537;=&#3627409361;&#3627408537;
&#3627409360;
−&#3627409362;&#3627408537;+&#3627409361;

Subtraction of Function:
Example 2:&#3627408520;&#3627408537;=−&#3627408537;
&#3627409360;
+&#3627409367;−&#3627409360;&#3627408537;
&#3627409361;
&#3627408514;&#3627408527;&#3627408517;&#3627408521;&#3627408537;=−&#3627409361;&#3627408537;+&#3627409362;
−&#3627409360;&#3627408537;
&#3627409361;
−&#3627408537;
&#3627409360;
+&#3627409367;(−&#3627409361;&#3627408537;+&#3627409362;)−
+&#3627409361;&#3627408537;−&#3627409362;
−&#3627409360;&#3627408537;
&#3627409361;
−&#3627408537;
&#3627409360;
+&#3627409361;&#3627408537;+&#3627409367;−&#3627409362;
&#3627408520;−&#3627408521;&#3627408537;=−&#3627409360;&#3627408537;
&#3627409361;
−&#3627408537;
&#3627409360;
+&#3627409361;&#3627408537;+&#3627409363;
−&#3627409360;&#3627408537;
&#3627409361;
−&#3627408537;
&#3627409360;
+&#3627409367;
−&#3627409360;&#3627408537;
&#3627409361;
−&#3627408537;
&#3627409360;
+&#3627409361;&#3627408537;+&#3627409363;

Subtraction of Function:
&#3627408520;&#3627408537;=−&#3627408537;
&#3627409360;
+&#3627409367;−&#3627409360;&#3627408537;
&#3627409361;
&#3627408514;&#3627408527;&#3627408517;&#3627408521;&#3627408537;=−&#3627409361;&#3627408537;+&#3627409362;
Using the same given in example 2, find:
(&#3627408520;−&#3627408521;)(&#3627409360;)
&#3627408520;−&#3627408521;&#3627408537;=−&#3627409360;&#3627408537;
&#3627409361;
−&#3627408537;
&#3627409360;
+&#3627409361;&#3627408537;+&#3627409363;
=−&#3627409360;(&#3627409360;)
&#3627409361;
−(&#3627409360;)
&#3627409360;
+&#3627409361;&#3627409360;+&#3627409363;
=−&#3627409360;&#3627409366;−(&#3627409362;)+&#3627409364;+&#3627409363;
=−&#3627409359;&#3627409364;−&#3627409362;+&#3627409359;&#3627409359;&#3627408520;−&#3627408521;&#3627409360;=−&#3627409367;

Multiplication of Function:
f*gxf x*gx
To find theproductoftwo functions, put
parenthesis around them and multiply
each term from the first function to each
term of the second function.

Multiplication of Function:
&#3627408519;&#3627408537;=−&#3627409365;&#3627408537;+&#3627409359;&#3627409360;&#3627408514;&#3627408527;&#3627408517;&#3627408520;&#3627408537;=&#3627409360;&#3627408537;−&#3627409363;Example 1:
−&#3627409365;&#3627408537;+&#3627409359;&#3627409360;
&#3627409360;&#3627408537;−&#3627409363;
∗
−&#3627409359;&#3627409362;&#3627408537;
&#3627409360;
+&#3627409360;&#3627409362;&#3627408537;
&#3627409361;&#3627409363;&#3627408537;−&#3627409364;&#3627409358;
−&#3627409359;&#3627409362;&#3627408537;
&#3627409360;
+&#3627409363;&#3627409367;&#3627408537;−&#3627409364;&#3627409358;
(−&#3627409365;&#3627408537;+&#3627409359;&#3627409360;)(&#3627409360;&#3627408537;−&#3627409363;)
??????=
??????=
??????=
??????=
−&#3627409359;&#3627409362;&#3627408537;
&#3627409360;
&#3627409361;&#3627409363;&#3627408537;
&#3627409360;&#3627409362;&#3627408537;
−&#3627409364;&#3627409358;
&#3627409363;&#3627409367;&#3627408537;
&#3627408519;∗&#3627408520;&#3627408537;=−&#3627409359;&#3627409362;&#3627408537;
&#3627409360;
+&#3627409363;&#3627409367;&#3627408537;−&#3627409364;&#3627409358;

Multiplication of Function:
Example 2:&#3627408532;&#3627408537;=&#3627409360;&#3627408537;
&#3627409360;
−&#3627409360;+&#3627409361;&#3627408537;
&#3627409361;
&#3627408514;&#3627408527;&#3627408517;&#3627408533;&#3627408537;=−&#3627408537;+&#3627409359;
&#3627409361;&#3627408537;
&#3627409361;
+&#3627409360;&#3627408537;
&#3627409360;
−&#3627409360;
−&#3627408537;+&#3627409359;
−&#3627409361;&#3627408537;
&#3627409362;
−&#3627409360;&#3627408537;
&#3627409361;
+&#3627409360;&#3627408537;
&#3627409361;&#3627408537;
&#3627409361;
+&#3627409360;&#3627408537;
&#3627409360;
−&#3627409360;
−&#3627409361;&#3627408537;
&#3627409362;
+&#3627408537;
&#3627409361;
+&#3627409360;&#3627408537;
&#3627409360;
+&#3627409360;&#3627408537;−&#3627409360;
&#3627408532;∗&#3627408533;&#3627408537;=−&#3627409361;&#3627408537;
&#3627409362;
+&#3627408537;
&#3627409361;
+&#3627409360;&#3627408537;
&#3627409360;
+&#3627409360;&#3627408537;−&#3627409360;

Multiplication of Function:
&#3627408532;&#3627408537;=&#3627409360;&#3627408537;
&#3627409360;
−&#3627409360;+&#3627409361;&#3627408537;
&#3627409361;
&#3627408514;&#3627408527;&#3627408517;&#3627408533;&#3627408537;=−&#3627408537;+&#3627409359;
Using the same given in example 2, find:
(&#3627408532;∗&#3627408533;)
&#3627409359;
&#3627409360;
&#3627408532;∗&#3627408533;&#3627408537;=−&#3627409361;&#3627408537;
&#3627409362;
+&#3627408537;
&#3627409361;
+&#3627409360;&#3627408537;
&#3627409360;
+&#3627409360;&#3627408537;−&#3627409360;
=−&#3627409361;
&#3627409359;
&#3627409360;
&#3627409362;
+
&#3627409359;
&#3627409360;
&#3627409361;
+&#3627409360;
&#3627409359;
&#3627409360;
&#3627409360;
+&#3627409360;
&#3627409359;
&#3627409360;
−&#3627409360;

Multiplication of Function:
=−&#3627409361;
&#3627409359;
&#3627409360;
&#3627409362;
+
&#3627409359;
&#3627409360;
&#3627409361;
+&#3627409360;
&#3627409359;
&#3627409360;
&#3627409360;
+&#3627409360;
&#3627409359;
&#3627409360;
−&#3627409360;
=−&#3627409361;
&#3627409359;
&#3627409359;&#3627409364;
+
&#3627409359;
&#3627409366;
+&#3627409360;
&#3627409359;
&#3627409362;
+&#3627409360;
&#3627409359;
&#3627409360;
−&#3627409360;
=
−&#3627409361;
&#3627409359;&#3627409364;
+
&#3627409359;
&#3627409366;
+
&#3627409359;
&#3627409360;
+&#3627409359;−&#3627409360;
&#3627408532;∗&#3627408533;
&#3627409359;
&#3627409360;
=
−&#3627409367;
&#3627409359;&#3627409364;

Multiplication of Function:
&#3627408532;∗&#3627408533;
&#3627409359;
&#3627409360;
=
−&#3627409367;
&#3627409359;&#3627409364;

Division of Function:
When you divide two such functions together,
you get what is called a rational expression.
A rational expression is the division of two
polynomials. If they divide evenly, your answer
will become a polynomial.

Division of Function:
Polynomial long-division
Synthetic division

Example 1:&#3627408519;&#3627408537;=&#3627409361;&#3627408537;
&#3627409360;
+&#3627409362;&#3627408537;+&#3627409363;&#3627408514;&#3627408527;&#3627408517;&#3627408520;&#3627408537;=&#3627408537;+&#3627409360;
&#3627409361;&#3627408537;
&#3627409360;
+&#3627409362;&#3627408537;+&#3627409363;&#3627408537;+&#3627409360;
&#3627409361;&#3627408537;
&#3627409361;&#3627408537;
&#3627409360;
−&#3627409364;&#3627408537;
−
−&#3627409360;&#3627408537;+&#3627409363;
−&#3627409360;
−
&#3627409360;&#3627408537;+&#3627409362;
&#3627409367;
&#3627408519;
&#3627408520;
&#3627408537;=&#3627409361;&#3627408537;−&#3627409360;+
&#3627409367;
&#3627408537;+&#3627409360;

Example:&#3627408519;&#3627408537;=&#3627409361;&#3627408537;
&#3627409360;
+&#3627409362;&#3627408537;+&#3627409363;&#3627408514;&#3627408527;&#3627408517;&#3627408520;&#3627408537;=&#3627408537;+&#3627409360;
??????&#3627408532;&#3627408522;&#3627408527;&#3627408520;??????&#3627408538;&#3627408527;&#3627408533;&#3627408521;&#3627408518;&#3627408533;&#3627408522;&#3627408516;&#3627408517;&#3627408522;??????&#3627408522;&#3627408532;&#3627408522;&#3627408528;&#3627408527;:
&#3627408537;+&#3627409360;=&#3627409358;
&#3627408537;+&#3627409360;−&#3627409360;=&#3627409358;−&#3627409360;
−&#3627409360;
&#3627408537;=−&#3627409360;
&#3627409361;&#3627409362;&#3627409363;
&#3627409361;
−&#3627409364;
−&#3627409360;
&#3627409362;
&#3627409367;
&#3627408519;
&#3627408520;
&#3627408537;=&#3627409361;&#3627408537;−&#3627409360;+
&#3627409367;
&#3627408537;+&#3627409360;
&#3627408531;&#3627408518;&#3627408526;&#3627408514;&#3627408522;&#3627408527;&#3627408517;&#3627408518;&#3627408531;

Division of Function:
Using the same given in the example, find:
&#3627408519;&#3627408537;=&#3627409361;&#3627408537;
&#3627409360;
+&#3627409362;&#3627408537;+&#3627409363;&#3627408514;&#3627408527;&#3627408517;&#3627408520;&#3627408537;=&#3627408537;+&#3627409360;
&#3627408519;
&#3627408520;
&#3627409359;
&#3627409360;
&#3627408519;
&#3627408520;
&#3627408537;=&#3627409361;&#3627408537;−&#3627409360;+
&#3627409367;
&#3627408537;+&#3627409360;
=&#3627409361;
&#3627409359;
&#3627409360;
−&#3627409360;+&#3627409367;
=
&#3627409361;
&#3627409360;
+&#3627409365;
&#3627408519;
&#3627408520;
&#3627409359;
&#3627409360;
=
&#3627409359;&#3627409365;
&#3627409360;
&#3627408528;&#3627408531;&#3627409366;
&#3627409359;
&#3627409360;

Composite Function:
Compositefunctionorcompositionof
functionisanotherwayofcombining
function.
Thismethodofcombiningfunctionusesthe
outputofonefunctionastheinputfora
secondfunction.

Composite Function:
f g xf [ g x]
This is read “f composition g” or “f composed
g” and means to copy the f function down but
where ever you see an x, substitute in the g
function.

Composite Function:
Example 1:&#3627408519;&#3627408537;=&#3627409362;&#3627408537;+&#3627409359;&#3627409358;&#3627408514;&#3627408527;&#3627408517;&#3627408520;&#3627408537;=&#3627408537;+&#3627409359;
=&#3627409362;(&#3627408537;+&#3627409359;)+&#3627409359;&#3627409358;
&#3627409362;&#3627408537;+&#3627409362;+&#3627409359;&#3627409358;
&#3627409362;&#3627408537;+&#3627409359;&#3627409362;
f[gx]=&#3627409362;&#3627408537;+&#3627409359;&#3627409362;

Composite Function:
Example 2:&#3627408521;&#3627408537;=&#3627409361;&#3627408537;
&#3627409360;
−&#3627408537;+&#3627409366;&#3627408514;&#3627408527;&#3627408517;??????&#3627408537;=−&#3627409360;&#3627408537;+&#3627409361;
&#3627409361;(−&#3627409360;&#3627408537;+&#3627409361;)
&#3627409360;
−(−&#3627409360;&#3627408537;+&#3627409361;)+&#3627409366;
&#3627409361;(&#3627409362;&#3627408537;
&#3627409360;
−&#3627409359;&#3627409360;&#3627408537;+&#3627409367;)+&#3627409360;&#3627408537;−&#3627409361;+&#3627409366;
&#3627409359;&#3627409360;&#3627408537;
&#3627409360;
−&#3627409361;&#3627409364;&#3627408537;+&#3627409360;&#3627409365;+&#3627409360;&#3627408537;−&#3627409361;+&#3627409366;
ℎ????????????=12??????
2
−34??????+32

Another one…
Given that:&#3627408519;&#3627408537;=&#3627409366;&#3627408537;+&#3627409360;&#3627408514;&#3627408527;&#3627408517;&#3627408520;&#3627408537;=−&#3627409361;&#3627408537;−&#3627409365;,&#3627408519;&#3627408522;&#3627408527;&#3627408517;:
&#3627409359;.&#3627408519;&#3627408520;&#3627408537;2.&#3627408519;&#3627408520;−&#3627409360;
&#3627409366;(−&#3627409361;&#3627408537;−&#3627409365;)+&#3627409360;
−&#3627409360;&#3627409362;&#3627408537;−&#3627409363;&#3627409364;+&#3627409360;
f[gx]=−&#3627409360;&#3627409362;&#3627408537;−&#3627409363;&#3627409362;
−&#3627409360;&#3627409362;&#3627408537;−&#3627409363;&#3627409362;
−&#3627409360;&#3627409362;(−&#3627409360;)−&#3627409363;&#3627409362;
&#3627409362;&#3627409366;−&#3627409363;&#3627409362;
&#3627408519;[&#3627408520;−&#3627409360;]=−&#3627409364;

Quotation of the day:
“ One of the lesson of math in our life
that we should to apply is always be
careful with the sign ”
-Anonymous

Operation of functions and Composite function.pdf

About This Presentation

Slide Content

Tags

Categories

Download

Quick Actions

Statistics

Related Slideshows

Operation of functions and Composite function.pdf

About This Presentation

Slide Content

Slide 1

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Tags

Categories

Download

Quick Actions

Statistics

Related Slideshows

Pray For The Peace Of Jerusalem and You Will Prosper

Don_t_Waste_Your_Life_God.....powerpoint

VILLASUR_FACTORS_TO_CONSIDER_IN_PLATING_SALAD_10-13.pdf

Fertility awareness methods for women in the society

Chapter 5 Arithmetic Functions Computer Organisation and Architecture

syakira bhasa inggris (1) (1).pptx.......