# Composition of Functions and Inverse of a Function

## Composite Function Definition

What is a composite function? Well, a composite function is usually composed of other functions such that the output of one function is the input of the other function. In other words, when the value of a function is found from two other given functions by applying one function to an independent variable and the other to the result of the other function whose domain consists of those values of the independent variable for which the result yielded by the first function lies in the domain of the second.

Example: Two functions – 3y+5 and y2 together forms a composite function which can be written as (3y+5)2

### Explanation of Composition Functions

To form a composite function by a composition of two other functions we need to take two functions say g(x) = $x^{2}$ and f(x) = x+5. Now, we need to put one function inside the other function so here we can put f(x) into g(x) to form a new function, called their composition.

As mentioned above, to form composite functions we need to insert one function into another. Here f(x) can be plugged into g(x) to form a function g(f(x)). We know that f(x) = x + 5, thus we can substitute the function in. Therefore, g(f(x)) = g(x + 5). Knowing the fact that g(x) = $x^{2}$ we can insert the function and evaluate g(x + 5) = $(x + 5)^{2}$. Therefore, g(f(x)) = g(x + 5) = $(x + 5)^{2}$ .

### Composite Functions Properties

There are four major properties of a composite function:

Property 1:  Composite functions are not commutative

gof is not equal to fog

Property 2: Composite functions are associative

(fog)oh = fo(goh)

Property 3:  A function f: A -B and g: B-C is one-one then gof: A-C is also one-one.

Property 4: A function f: A-B and g: B-C is onto then gof: A-C is also onto.

### What is Inverse Function?

An inverse function is a function, which can reverse into another function. In other words, if any function “f” takes p to q then, the inverse of “f” i.e. “f-1” will take q to p.  A function accepts a value followed by performing particular operations on these values to generate an output. If you consider functions, f and g are inverse, then f(g(x)) is equal to g(f(x)) which is equal to x.

Given below are the detailed summary of the Composition and inverse relation with examples:

## Composite and Inverse Functions

 Function Inverse Function Composite Function Definition In mathematics, a function, a, is defined as an inverse of another, b, if the output of b is given, a, returns the input value that was given to b. Also, it must be true for every element in the domain as well as the co-domain(range) of b. In other words, assuming p and q are constants if b(p) = q and a(q) = p then the function a will be called an inverse of the function b. A function whose input is another function is called a composite function.. So, if we have two functions A(x), which draws elements from set B to set C, and D(x), which draws from set C to set E, then the composite of these two functions, will be written as DoA, which is a function that draws elements from B to E i.e. DoA is equal to D(A(x)). Examples Example of Inverse Function – Consider the functions a(x) = 5x + 2 and b(y) = (y-2)/5. Here function b is an inverse function of a.This is visible by inserting values into the functions. For example when x is 1 the output of a is a(1) = 5(1) + 2 = 7. Using this output as y in function b gives b(7) = (7-2)/5 = 1 which was the input value to function a. For example consider the functions A(x) = 5x + 2 and B(x) = x + 1. The composite function AoB = A(B(x)) = 5(x+1) + 2. Properties Listed below are some of the properties of Inverse Functions: Two functions f and g will be referred to as an inverse of each other if: Both f and g are one to one functions. In one to one functions, each value is mapped in their domain to exactly one value in the co-domain(range). Here is an example of a One to One function: f(x) = x The co-domain(range) of one function(f) is the domain of another function(g) and vice versa Note: Some functions are invertible only for a set of specific values in their domain. By chance, if both the range as well as the domain of the inverse function are restricted to only those values. Listed below are some properties of Composite Functions: Composite functions consist of the following properties: Given that the composite function is fog = f(g(x)) the co-domain of g has to be a subset, i.e. either proper or improper subset, of the domain of f Composite functions are always associative. Given that the composite function is a o b o c then the order of operation will be irrelevant i.e. (a o b) o c is equal to a o (b o c). Composite functions  cannot be commutative. So AoB is not equal  to BoA. Using the example A(x) = 5x + 2 and B(x) = x + 1 AoB = A(B(x)) = 5(x+1) + 2 while BoA = B(A(x)) = (5x + 2) + 1.

Solved Examples

Question 1) Let f(x) = $x^{2}$ and g(x) = $\sqrt{1 – x^{2}}$ Find (gof)(x) and (fog)(x).

Solution 1) (gof)(x) = g(f(x)) = g($x^{2}$) = $\sqrt{1 – (x^{2})^{2} = \sqrt{1 – x^{4}}}$

(fog) (x) = f(g(x)) = f ($\sqrt{1 -x^{2})}$ = 1 -$(x^{2})^{2}$ = 1 – $x^{2}$

Question 2)  If f(x) =$x^{2}$ , g(x) = $\frac{x}{3}$  and h(x) = 3x+2 . Find out fohog(x).

Solution 2) h(g(x)) = 3 $\left ( \frac{x}{3} \right )$ + 2 = x + 2

fohog(x) = f [h(g(x))]  = $(x + 2)^{2}$

Therefore this is the required solution.