Analysis of Composite and Inverse Functions: Examples and Applications

This assignment explores the concepts of composite and inverse functions, providing definitions and examples to illustrate their properties. The student reflects on these functions, offering an example of a composite function and explaining the process of finding the composite function. Inverse composite functions are also discussed, along with a real-world example of their occurrence in a grocery store. The assignment also covers strategies for graphing composite and inverse functions, such as using Desmos calculators, and discusses the concepts of domain and range in relation to these functions. The student references relevant literature and demonstrates a solid understanding of the subject matter.

Running head : COMPOSITE FUNCTIONS
COMPOSITE FUNCTIONS
Name of Student
Institution Affiliation

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.

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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.