[Maths Class Notes] on Introduction to the Composition of Functions and Inverse of a Function Pdf for Exam

Composition of a function and its inverse are two mathematical concepts with practical applicability. The objective of these two concepts is to increase the understanding of functions and all the terms related to it. Students will better understand the definitions of domain and range after going through these two concepts.

The objective of the composition of functions and inverse of a function is to develop an application based thinking of how the functions work. Both of these concepts have a real-life application. Students are advised to regularly give time and effort to mathematics and increase their score in it.

Composite Functions and Inverse Functions

Let us try and understand both of these mathematical concepts in detail. Composition of functions and inverse functions are easy concepts to understand and apply. Students are advised to try as many examples as possible to solidify their learning and understanding of both the concepts. Given below is the detailed explanation of both the concepts: 

Composition of Functions

Composition of function is defined when the result of a function is obtained by applying another function. The independent variable is another function. Let us try to understand the composition of functions with the help of an example.

Let there be two functions, f(x) and g(x).

f(x) = 2x + 1 g(x)=x2

Let us find the value of g(x) with the result obtained from f(x).

Calculate f(x) at 1.

f(1)=2.1+1=3

Let us calculate g(x) at 3.

g(3)=3²= 9.

To streamline the above process and understand it better, we create a new function. This is how we represent the composition of functions.

f(g(x))=(fog)(x)

Where o is the composition operator and is used to define the composition of functions. Let us try and solve the above problem with this representation at x=1.

(gof)(x)=g(f(x))

=g(2x+1) 

=(2x+1)2

=4x2+ 4x + 1

= 9

The notation (fog) is read as f of g or f is composed of g.

Inverse Functions

Inverse functions, as the name suggests, is to describe an inverse relationship between two functions. The two functions are opposite of each other. Let us try and understand this concept using a common example.

Let us take the case of temperature scales. Two scales, degree Celsius scale and the Fahrenheit scale, are used to measuring temperature.

C(x)=5/9(x-32) converts fahrenheit to degree celsius.

F(x)=9/5(x) + 32 converts degree celsius into degree fahrenheit. 

We wish to convert 77oF into degree celsius.

C(x)=5/9(77-32)

=5/9(45)

=5.5

=25oC

Now to understand the inverse function, let us convert this degree celsius into degree fahrenheit.

F(x)=9/5(x)+32

=9/5(25)+32

=9.5+32

=45+32

=77oF

We get the same Fahrenheit value we began with. Inverse functions are opposite of each other. We can obtain any of the functions.

C(x) = 5/9(x-32)

9/5(C(x)) = x-32

9/5(x) + 32 = F(x)

This is how the inverse functions work.