In mathematics, a function is a specific relationship between inputs (the domain) and outputs (the co-domain). Each input has exactly one output, which can again be traced back to its input.

There are several types of functions in maths, such as:

• • One-to-one or injective function: The domain of a function is considered a one-to-one function if each element in the domain has a distinct image in the co-domain. There is a mapping between two sets for a range in each domain.

  • Many-to-one or not injective function: The domain of a function is considered a many-to-one function if there is at least one element in the domain that does not have a distinct image in the co-domain. 
  • Surjective or onto function: When all the elements in a co-domain are mapped from a domain, that is, the co-domain is equal to the range, it is a surjective function.
  • Injective function: When the co-domain is not equal to the range, it is an injective function.

• One-One mapping/function or Injective Mapping:

A function f: P→Q is said tobe one-one (or injective) if different elements of P have different images in Q. We can also say that, if a function f: P→Q is such that f(x)=f(y)=> x=y

Or, x ≠ y => f(x)≠ f(y) for all x,y ∈ P, then this is called a one-one function, injective mapping or Injection. 

It is readily followed that, if P and Q are two finite sets, then the function f: P→Q is an injection, when n(P)<=n(Q)

Ex:

 Is f (x) = 8x – 7 one-one where f : R→R.

The above given function is one-one. Because this is a straight line curve that possesses one output for a single input. 

 Is f (x) = | x – 8 | one-one where f : R→R

This function is not one-one because it shows the same output for different values of input.  

Composite functions

When a function is dependent on another, it is called a Composite Function; it is created when both of them are merged together.

How can you solve composite functions?

Solving a composite function means locating the structure of two features. We use a little circle (∘) because of the construction of a characteristic. Here are the steps to solving a composite function: 

Rewrite the composition in an alternative way.

For example, 

(f ∘ g) (x) = f [g (x)]

(g ∘ f) (x) = g [f (x)]

(f ∘ g) (x²) = f [g (x²)]

Properties of composite functions

Suppose there are two functions: the first is bijective, and the second is injective. What could be said about the composition beforehand (i.e. without checking out the accurate or actual composition)?

The feature composition of the one-to-one function is one to one. 

Associative property of Composite function

Let’s find out how to prove the function structure is associative. Let A, B, C, and D be sets and

 f: C D, g: BC, and h: A B. Showing that fo(goh) = (fog)oh:

Let f: X→Y, g: Y→Z, h: Z→W be functions

((𝑓∘𝑔)∘ℎ)(𝑥) = 𝑓∘𝑔(ℎ(𝑥)) = 𝑓(𝑔(ℎ(𝑥))

(𝑓∘(𝑔∘ℎ))(𝑥) = 𝑓(𝑔∘ℎ(𝑥)) = 𝑓(𝑔(ℎ(𝑥)).

Commutative property of Composite function

The order of some mathematical activities is the topic of the commutative property. For example, the situation a x b = b x a may be applied to show the operation, which involves just two elements. The order of the components doesn’t impact the outcome of the processes. However, the same cannot be applied to the composition of functions:

 g ∘ f f ∘ g (may be equal in some specific cases only)

A few more properties:

• A one-to-one function’s function composition is always one to one.
• When two onto functions are combined, the result is always onto.
• The inverse of the composition of two functions f and g is the inverse of the composition of both functions, such as (f ∘ g)-1= ( g-1 ∘ f-1).

Examples of composition of functions

1. Emily and her family went to an amusement park. The entry tickets were priced differently for people of various ages. The age groups were divided into children, adults, and senior persons. What is the best way to connect the given data to composite functions?

Solution: 

We use the function f(y) = ticket cost.

g(x) = Person’s age group is another function.

We can say that since the price of a ticket is determined by the person’s age group,

y = g(x).

As a result, the composite function to describe the cost of a ticket is as follows:

(f(g(x))).

(f(g(x)) is the general form of a composite function.

2. Given that f(x) = 2x and g(x) = x+1, find (f∘g)(x) if x = 1.

Conclusion

The composition of functions is an essential topic. The composition of functions is an operation where two functions, say F(x) and G(x), make a new function H(x) in a way where H(x) = F(G(x)). 

Here G(x) is applied to x, and the value obtained is put in F(x) to get the final value.

On the other hand, a one-one function is the function if each element in the domain has a distinct image in the co-domain.