Composite Functions | Assignment

Running head : COMPOSITE FUNCTIONS
COMPOSITE FUNCTIONS
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COMPOSITE FUNCTIONS 2
Reflect on the concept of composite and inverse functions
Composite function
Let f : A B and g: B C to be two functions . The function f takes an element x ∈A to an
element f(x) = y ∈B and y is further taken by g to an element Z ∈C Therefore Z=g(y) = g of n
then for each x∈A a unique element Z=g(f(x)) of C (Cao, 2011) . This rule is therefore a
function of f and g´ . Then if f:A B and g: B C be any two function then the composite
function of f of g is denoted by g of f and it is a function of g of A C is defined by ( g of
) ( x) = g(f(x)) ∀ x ∈ A
Inverse function
Let f be a 1-1 function from A onto B. Therefore ∀ yϵB there exist x∈A such that f (x) = y and
since y is a 1-1 function (Shapiro, 2012). We can define a function say g from B onto A such that
g(y) = x. This function g is therefore called inverse function of f and denoted by function f-1.
What are the simplest composite and inverse functions you can imagine?
Example of composite function
Given f(x) = x2 + 6 and g(x) = 2x – 1, find
(f ο g)(x)
(f ο g)(x)
= f(2x – 1)
= (2x – 1)2 + 6
= 4x2 – 4x + 1 + 6
= 4x2 – 4x + 7
Inverse composite functions
As an example, consider

COMPOSITE FUNCTIONS 3
f(x)=3+2x⟺f−1(x)=x−32f(x)=3+2x⟺f−1(x)=x−32
.
Then
(f∘f−1)(x)=f(f−1(x))=f(x−32)=3+2x−32=x(f∘f−1)(x)=f(f−1(x))=f(x−32)=3+2x−32=x
And (f−1∘ f)(x)=f−1(f(x))=f−1(3+2x)=3+2x−32=x
In your day to day, is there any occurring fact that can be interpreted as composite and
inverse functions
Yes in a grocery. Let A be the set of fruits in a given store. There is a function f:A R+ that
maps a fruit to its weight in kilogram. And also a function g: R+ R+ which converst a
kilogram weight to tones the same weight measured in tons.
Strategy are you using to get the graph of composite and inverse functions?
A good strategy which can be employed in plotting these function is to use desmos calculators ,
for example typing y=x3 for a range of -2 , x< 2¿, y = x1/3 for a range of -2¿ x <2¿ and y = x -2
¿ x <2¿ then describe the relationship between the three curves. Suppose f = R R is a function
from the set of real numbers to set with f(x) = x+1 we write f2 to represent f o f and f n+1 = fn o f .
It is true that f2 o f = f 0 f2.
What concepts (only the names) did you need to accommodate these functions
1. Domain . Domain ( g o f ) = domain in f of codemain ( g o f ) = codomain ( g).
2. Range . g o f is well defined only if range S is long.

COMPOSITE FUNCTIONS 4
References
Cao, J., Lin, Z., & Huang, G. B. (2011). Composite function wavelet neural networks with differential
evolution and extreme learning machine. Neural Processing Letters, 33(3), 251.
Shapiro, J. H. (2012). Composition operators: and classical function theory. Springer Science & Business
Media.