# 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) = \[(x + 5)^{2}\] .

For practice, download composition of functions examples with answers pdf. By downloading composition of functions examples with answers pdf, you will have enough composite functions questions for practising.

### 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

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.

1. What is the difference between one to one function and one to many functions?

Functions can be described as something which gives output to an input. It might be possible that it gives the same output to more than one input but it won’t be called a function if it gives two outputs for a single input. A function will be one to one if no two inputs have the same output. Or else, if we can find at least more than one input for which function has the same output then it’s a many to one function. One to one function is an important criteria for a function to be reversible. There are numerous ways to check if a function is one to one function or many to one function. It is preferable to solve it using a graph. A continuous monotonic function is always one to one whereas a continuous non-monotonic function is always many to one.

2. Why do we have to find an inverse of a function? Is there any physical significance of using inverse?

The use of inverse function makes problem-solving much easier. For example, calculation in certain areas in calculus. There are cases where we will have only 1 double integral if we use the inverse of the functions, instead of 2 double integrals if we use the functions directly.

This stands true in the field of economics also. We can define certain concepts belonging to economics easily using the inverse demand curve instead of the demand curve.