Composition of Functions: Representation, Process of Solving It and Symbol

To understand the composition of functions, we will consider an example:

Let A, B and C be three non-empty sets. Let f: A $\to$ B and g: B $\to$ C be two functions. Since, f is a function from A to B, for every x $\in$ A, there is a unique element f(x) $\in$ B.

Since, g is a function from B to C, for every f(x) $\in$ B, there is a unique element g(f(x)) $\in$ C. i.e., for every x $\in$ A, there  is a unique element g(f(x)) $\in$ C.

In other words, we have a new function from A to C. This new function is called composition of f and g and we denote this function by gof.

Hence, gof: A $\to$ C is defined by (gof)(x) = g(f(x)).

Students can practice NCERT solutions for relations and functions to understand the types of questions asked in the CBSE exam.

Introduction:

An inverse function is a function that "undoes" another function. It takes the output of the original function and returns the original input. Essentially, an inverse function swaps the input and output values of the original function. JEE Main and IIT JAM are some of the competitive exams held in the country which ask questions based on the topic.

Definition:

Let A and B be two non-empty sets. Let f: A $\to$ B be a bijection.

Let g: B $\to$ A be a function which associates each element y $\in$ B to a unique element x $\in$ A

(i.e., g(y) = x) such that f(x) =y, then g is called the inverse function of f or inverse of f.

We denote the inverse function of f by f^-1 (i.e., g = f^-1).

In this case, we have f(x) = y $\iff$ x = g(y)

A function whose inverse exists is called invertible function or inversible function.

Thus, if f: A $\to$ B be a bijection, then f^-1: B $\to$ A is such that

f(x) = y $\iff$ x = f^-1(y) 

1. First, you need to identify which function will be the "inner" function and which will be the "outer" function. This is an important for aspirants of NEET and IISER exams.
2. Then, substitute x into the inner function and try to simplify it as much as possible.
3. Substitute the result from step 2 into the outer function.
4. Combine similar parts and simplify the last and final expression.

Aspirants of CUET and other competitive exams must keep the following pointers in point to perform well:

• The composition gof exists, if the Range of f $\subseteq$ Domain of g.
• The composition fog exists, if the Range of g $\subseteq$ Domain of f.
• It may be possible that gof exists but fog does not exist.
• gof and fog may or may not be equal.
• Only bijective functions have inverse.
• Inverse of a function, if exists, is unique.
• If f: A $\to$ B is a bijection, then f^-1o f = I_A and f o f^-1 = I_B, where I_A and I_B denote identity functions on A and B respectively.
• If f: A $\to$ B is a bijection, then f^-1: B $\to$ A is also bijection and (f^-1)^-1 = f.
• If f: A $\to$ B and g: B $\to$ C are bijections, then gof is also bijection (gof)^-1 = f^-1 o g^-1.
• If f: A $\to$ B and g: A $\to$ A be two functions such that gof = I_A = fog, then f and g are bijective functions and g = f^-1.