# Composition of Functions

Definition: Let f: A → B and g: B → C be two functions. Then a function gof: A → C defined by (gof)(x) = g(f(x)), for all x ϵ A is called the composition of f and g. Note 1: It is evident from the definition that gof is defined only if for each x ϵ A, f (x) is an element of g so that we can take its g – image. Hence, for the composition gof to exist, the range of f must be a subset of the domain of g.

Note 2: It should be noted that gof exists if the range of f is a subset of domain of g. similarly, fog exists if range of g is a subset of domain of f.

Example: Let f: R → R; f (x) = sin x and g : R → R, g (x) = x² find fog and gof.

Solution: Clearly, fog and gof both exist.

Now, (gof)(x) = g(f(x)) = g(sinx) = (sinx)² = sin²x

And, (fog)(x) = f(g(x)) = f(x²) = sinx².

Example: Let f(x) =\left\{ \begin{align}& 1+x,\,\,\,\,\,0\le x\le2 \\& 3-x,\,\,\,\,\,2<x\le 3 \\\end{align} \right. find fof (x).

Solution: We have, fof (x) = f(f(x))

x =\left\{ \begin{align}& f\left( 1+x \right),\,\,\,\,\,0\le x\le 2 \\& f\left( 3-x\right),\,\,\,\,2<x\le 3 \\\end{align} \right.

=\left\{ \begin{align}& f\left( 1+x \right),\,\,\,\,\,0\le x<1 \\& f\left( 1+x \right),\,\,\,\,\,1<x\le 2 \\& f\left( 3-x \right),\,\,\,\,2<x\le 3 \\\end{align} \right..

=\left\{ \begin{align}& 1+\left( 1+x \right),\,\,\,\,\,0\le x<1 \\ & 3-\left( 1+x \right),\,\,\,\,\,1<x\le 2 \\& 1+\left( 3-x \right),\,\,\,\,2<x\le 3 \\\end{align} \right..

=\left\{ \begin{align}& 2+x,\,\,\,\,\,0\le x\le 1 \\& 2-x,\,\,\,\,\,1<x\le 2 \\& 4-x,\,\,\,\,2<x\le 3 \\\end{align} \right..

PROPERTIES OF COMPOSITION OF FUNCTIONS:

Property 1: The composition of functions is not commutative i.e., fog ≠ gof.

Property 2: The composition of functions is associative i.e., if f, g, h are three functions such that (fog)oh and fo(goh) exist, then (fog)oh = fo(goh).

Property 3: The composition of two bijections is a bijection i.e., if f and g are two bijections, then gof is also a bijection.

Property 4: Let f : A → B. Then foIA = IBof = f i.e., the composition of any function with the identity function is the function itself.

Inverse of an Function: If f : A → B is a bijection, we can define a new function from B to A which associates each element y ϵ B to its pre-image f¯¹ (y) ϵ A. Such a function is known as the inverse of function f and is denoted by f¯¹. Example: If f : R → R is a bijection given by f (x) = x³ + 3, find f¯¹ (x).

Solution: Let f (x) = y then, f (x) = y

⇒ x³ + 3 = y

⇒ x = (y – 3)

⇒ f¯¹ (y) = (y – 3)

Thus, f¯¹ : R → R is given by f¯¹ (x) = (x – 3) for all x ϵ R.

PROPERTIES OF INVERSE OF A FUNCTION:

Property 1: The inverse of a bijection is unique.

Property 2: The inverse of a bijection is also a bijection.

Property 3: If f : A → B is a bijection and g : B → A is the inverse of f then fog = IB and gof = IA, Where IA and IB are the identity functions on the sets A and B respectively.

Property 4: If f : A → B and g : B → C are two bijections, then gof : A → C is a bijection and (gof)¯¹ = f¯¹og¯¹

Property 5: Let f : A → B and g : A → B be two functions such that gof = IA and fog = IB. Then, f and g are bijections and g = f¯¹.

Example: Show that the function f : R → R given by $$f\left( x \right)={{\log }_{a}}\left( x+\sqrt{{{x}^{2}}+1} \right)$$, a > 0, a ≠ 1 is invertible and find its inverse.

Solution: In order to prove that f (x) is invertible, it is sufficient to show that f is a bijection.

f is an injection: Let x, y be any two distinct real numbers. Then, x ≠ y.

⇒ $$x+\sqrt{{{x}^{2}}+1}\ne y+\sqrt{{{y}^{2}}+1}$$.

⇒ $${{\log }_{a}}\left( x+\sqrt{{{x}^{2}}+1} \right)\ne {{\log }_{a}}\left( y+\sqrt{{{y}^{2}}+1} \right)$$.

⇒ f (x) ≠ f (y)

⇒ f is an injection.

f is a surjection: Let f (x) = y Then, $${{\log }_{a}}\left( x+\sqrt{{{x}^{2}}+1} \right)=y$$.

$$x+\sqrt{{{x}^{2}}+1}={{a}^{y}}$$.

$$-x+\sqrt{{{x}^{2}}+1}={{a}^{-y}}$$.

∴ 2x = (ay – a-y)

$$x=\left( \frac{{{a}^{y}}-{{a}^{-y}}}{2} \right)$$.

Thus, for every y ϵ R, there exists $$x=\frac{{{a}^{y}}-{{a}^{-y}}}{2}\in R$$ such that f (x) = y. Therefore, f is a surjection.

Hence f is a bijection. Consequently, it is invertible.

In order to find f ¯¹ let f (x) = y

⇒ $${{\log }_{a}}\left( x+\sqrt{{{x}^{2}}+1} \right)=y$$.

⇒ $$x+\sqrt{{{x}^{2}}+1}={{a}^{y}}$$.

⇒ $$-x+\sqrt{{{x}^{2}}+1}={{a}^{-y}}$$.

∴ 2x = (ay – a-y)

⇒ x = ½ (ay – a-y)

⇒ f ¯¹ = ½ (ay – a-y)

Hence, f ¯¹ (x) = ½ (ax – a-x)