## Introduction

In mathematics, function composition is the pointwise application of one function to the results of another to produce a third function. That is, for functions f and g, function composition is defined as h(x) = g(f(x)). For instance, if f is a function that doubles its input and g is a function that adds 5 to its input, then the composition of these functions (denoted as g ∘ f) would be a function that takes an input, doubles it, and then adds 5. So, if we input the number 3 into g ∘ f , it would first double 3 to make 6 , and then add 5 to make 11 .

The composition of functions is a binary operation{\displaystyle \circ }\circ on the set of all functions from X to Y . The result of applying the composition of two functions f and g , denoted as {\displaystyle (f\circ g)(x)} (f\circ g)(x) or {\displaystyle f\circ g}f\circ g , is another function from X to Y . It is defined as the function h such that h(x) = g(f(x)) for all x in X . So we can say that 1) The domain of h is equal to the intersection of the domains of f and g . 2) The codomain of h is equal to the codomain of g .

The study of behavior under repeated applications of a given function goes back at least as far as Isaac Newton’s work on power series; see NEWTON’S METHOD. More generally, iteration theory studies iterates f^{n}(a), where n ranges over natural numbers or more generally over any semigroup. If we allow iteration in both directions (i.e., define b^{-n} recursively by b^{-n} := (b^{-1})^n ), this theory becomes invertible iteration theory.

## What is the inverse function?

The inverse function is a function that “undoes” another function. For example, the inverse of the function f(x) = 2x+1 is the function g(x) = (x-1)/2, which undoes the work of f(x). When we say that g(x) is the inverse of f(x), we usually write g = f-1.

## How to express the function in the form f g h use nonidentity functions for f g and h

To express the function in the form f g h, we need to use nonidentity functions for f, g, and h. We can do this by choosing different values for the inputs and outputs of each function. For example, let’s say we have a function that takes in a value x and outputs a value y. If we want to express this function in the form f g h, we could choose to input the values 1, 2, and 3 into the function, and output the values 4, 5, and 6 respectively. This would give us the following functions:

f(x) = 4

g(x) = 5

h(x) = 6

Now, let’s say we want to express the same function in the form h g f. In this case, we would need to input the values 3, 2, and 1 into the function, and output the values 6, 5, and 4 respectively. This would give us the following functions:

h(x) = 6

g(x) = 5

f(x) = 4

## Conclusion

When you’re trying to express a function in the form f g h, it’s important to use nonidentity functions for f, g, and h. This will help to prevent any confusion or errors in your work.