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Function is a relation on two sets by a rule. It is a special mapping between two sets. It emerges that it is possible to combine two functions, provided co-domain of one function is domain of another function. The composite function is a relation by a new rule between sets, which are not common to the functions.

We can understand composition in terms of two functions. Let there be two functions defined as :

f : A B by f(x) for all x A

g : B C by g(x) for all x B

Observe that set “B” is common to two functions. The rules of the functions are given by “f(x)” and “g(x)” respectively. Our objective here is to define a new function h : A C and its rule.

Thinking in terms of relation, “A” and “B” are the domain and co-domain of the function “f”. It means that every element “x” of “A” has an image “f(x)” in “B”.

Similarly, thinking in terms of relation, “B” and “C” are the domain and co-domain of the function “g”. In this function, "f(x)" – which was the image of pre-image “x” in “A” – is now pre-image for the function “g”. There is a corresponding unique image in set "C". Following the symbolic notation, "f(x)" has image denoted by "g(f(x))" in "C". The figure here depicts the relationship among three sets via two functions (relations) and the combination function.

Composition of two functions

Composition functions is a special relation between sets not common to two functions.

For every element, “x” in “A”, there exists an element f(x) in set “B”. This is the requirement of function “f” by definition. For every element “f(x)” in “B”, there exists an element g(f(x)) in set “B”. This is the requirement of function “g” by definition. It follows, then, that for every element “x” in “A”, there exists an element g(f(x)) in set “C”. This concluding statement is definition of a new function :

h : A C by g(f(x)) for all x A

By convention, we call this new function as “gof” and is read as "g circle f" or "g composed with f".

g o f x = g f x for all x A

The two symbolical representations are equivalent.


Problem 1: Let two sets be defined as :

h : R R by x 2 for all x R

k : R R by x + 1 for all x R

Determine “hok” and “koh”.

Solution : According to definition,

h o k x = h k x

h o k x = h x + 1

h o k x = x + 1 2

Again, according to definition,

k o h x = k h x

k o h x = k x 2

k o h x = x 2 + 1

Importantly note that h o k x k o h x . It indicates that composition of functions is not commutative.

Existence of composition set

In accordance with the definition of function, “f”, the range of “f” is a subset of its co-domain “B”. But, set “B” is the domain of function “g” such that there exists image g(f(x)) in “C” for every “x” in “A”. This means that range of “f” is subset of domain of “g” :

Range of “f” Domain of “g”

Clearly, if this condition is met, then composition “gof” exists. Following this conclusion, “fog” will exist, if

Range of “g” Domain of “f”

And, if both conditions are met simultaneously, then we can conclude that both “gof” and “fog” exist. Such possibility is generally met when all sets involved are set of real numbers, “R”.


Problem 2: Let two functions be defined as :

Questions & Answers

how do you translate this in Algebraic Expressions
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
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Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
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In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
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I'm interested in Nanotube
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Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
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Smarajit Reply
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Period of sin^6 3x+ cos^6 3x
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Period of sin^6 3x+ cos^6 3x
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Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
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