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

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.

FAQ (Frequently Asked Questions)

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.