# Composition of Functions & Inverse of a Function

## Composite Functions

When two functions are combined in such a way that the output of one function becomes the input to another function, then this is referred to as composite function.

Consider three sets X, Y and Z and let *f : X → Y* and g: Y → Z.

According to this, under map f, an element x ∈ X is mapped to an element y = f(x) ∈ Y which in turn is mapped by g to an element z ∈ Z in such a manner that z = g(y) = g[(*f*(x)] .

This mapping comprising of mappings f and g is known as **composition of mappings**. It is denoted by go*f* . Therefore, we are mapping onto .

The composite function is denoted as:

\(~~~~~~~~~\) ( gof)(x) = g(*f (X) )*

Similarly, (*f*og) (x) = *f* (g(x))

So, to find (gof) (x), take f(x) as argument for the function g.

Let us try to solve some questions based on composite functions.

Using Binomial Expansion, we have \((gof) (x) = 2 \left [ (3x)^{3} + 3.(3x)^{2}.5 + 3.(3x)(5)^{2}+(5)^3 \right ]\) \(\Rightarrow (gof) (x) = 2 \left [ 27x^{3} + 135x^{2} + 225 x + 125 \right]\) \(\Rightarrow (gof) (x) = 54x^{3} + 270x^{2} + 450 x + 250 \) Now, \(\Rightarrow (fog)(x) = 6x^{3} + 5\)
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## Inverse of a Function

Let *f*:X → *Y*. Now, let *f* represent a one to one function and y be any element of *Y* , there exists a unique element x ∈ *X* such that y = *f*(x).Then the map

\(~~~~~~~~~~\) \( f^{-1}:f[X] \rightarrow X \)

which associates to each element is called as the inverse map of *f*.

The function f(x) = \( x^5 \) and g(x) = \( x^{\frac 12} \) have the following property:

\( f\left( g(x) \right) \) = \( f \left( x^{\frac 15} \right) \) = \( (x^{\frac 15} )^5 \) = x

\( g\left( f(x) \right) \) = \( g \left( x^{5} \right) \) = \( (x^{5} )^{\frac 15} \) = x

Thus, if two functions *f* and g satisfy \( f \left( g(x) \right) \) = x for every x in domain of *f* , then in such a situation we can say that the function *f* is the inverse of g and g is the inverse of *f* .

For finding the inverse of a function,we write down the function y as a function of x i.e. y =* f*(x) and then solve for x as a function of y.

To have a better insight on the topic let us go through some examples.

This is the required solution.
To find the inverse, we need to write down this function as \(~~~~~~~~~~~~~\) y = \( x^3 \) In the above equation,y is an arbitrary element from the range of \(~~~~~~~~~~~~~\) x = \( y^{\frac 13} \) This gives a function g:Y →X. This new function g can be defined as \(~~~~~~~~~~~~~\) g(y) = \( y^{\frac 13} \) This function g is the inverse of the function |