Differential Composite Function

To begin with, a composite function can be understood as a function of a function, in essence, a function whose domain comprises the co-domain of another function. Such functions carry the properties of their constituent functions. If there exist two functions, F(Y)=Z and G(X)=Y, then the composite function FoG(X) will be a one-to-one function if both F and G are one-to-one functions. It is important here to recall that one-to-one functions are those functions in which each element of the domain of the function has exactly one image in the co-domain of the function. 

Understanding Composite Functions

Let us understand the concept of composite functions with the help of a simple example.

Consider a function F mapped from X to Y meaning that this function shows a relationship between elements of domain X with those of co-domain Y, hence F(X)=Y. There is another function G mapped from Y to Z, with similar connotations, implying G(Y)=Z. Then we can define a composite function GoF will be one mapped from X to Z where the domain of the composite function is the domain of the inner function and the range of the composite function is the range of the outer function. 

We can denote this as G[F(X)] = Z.

It is important to note here that composite functions can be associative in the sense that F(GoH) is the same as FoG(H) where F, G, and H are three different functions. However, it cannot be the case that FoG and GoF are used interchangeably. 

Differentiating Composite Functions

Consider two functions: F(x)=x2 and G(y)=y+3.

Then FoG can be written as FoG(y)=(y+3)2. This function can easily be expanded as FoG(y)=y2+6y+9. It can then be differentiated to give d F0G/ dy = 2y+6.

But if the functions are not as simple as quadratic functions, then this method can be quite tedious. Say if the function F changes to F(x)=x17 while the function G remains the same, then it is not feasible to expand the entire function and then take its derivative. 

In this case, we can differentiate FoG(y) by first differentiating the ‘outer’ function that is F(x)=x17 here and then multiplying it with the differential of the ‘inner’ function, G(y)=y+3. This is what is generally defined as the chain rule. 

In conclusion, d FoG/dy = 17x(1) = 17x where the value of x can vary.

Alternatively, if y is a function of u and u is a function of x then we can write: dy/dx = dy/du * du/dx. It is crucial to remember that dy/du and du/dx are not fractions here, rather they are differentials of y and u with respect to u and x, respectively. 

Determining Injectivity of a Function using Differentials

We begin this segment by understanding the concept of injectivity. An injective function is essentially a one-to-one function meaning that each element in the domain of the function has a unique image in the co-domain of the function.

A continuous and differentiable function whose differential is strictly increasing (dy/dx>0) or strictly decreasing (dy/dx<0) is a one-to-one function. This can be understood through the following analogy.

We know that the Mean Value Theorem states that a function if continuous over [a,b] and differentiable over (a,b), then there exists at least one point c belonging to the interval (a,b) such that f’(c) = f(b)-f(a) / b-a.

If any two numbers x1 and x2 in the interval [a,b] which violate the one-to-one function property such that x1 is not equal to x2 but f(x1) = f(x2), then there must exist c such that f’(c) = 0. Consequently, if a function is strictly increasing or strictly decreasing, then there cannot exist two numbers such as x1 and x2 which would violate the condition of a one-to-one function making the function one-to-one.

Inverse Functions

An inverse function of say function F(x)=y is a function G(y)=x. Conventionally, we write G(y) as F-1(x). The composite function of the initial function F(x) and its inverse F-1(x) gives and identity function that is F-1(F(x))=x and conversely F(F-1(x))=x.

In this case, it is necessary that the inner function of this composite function is a one-to-one function and an onto function, failing which the inverse of this function cannot exist.

Conclusion

Composite functions are thus combinations of more than one functions that form a unique function of their own. The properties of the constituent functions impact the properties of the composite functions. We can note here that if there are two functions F and G where F is one-to-one but G is not then FoG or Gof will not be a one-to-one function. Through our understanding of increasing and decreasing functions, we have also concluded that a differentiable (and thus a continuous) function which is strictly increasing or decreasing is a one-to-one function.