Composition of Functions

Composite Function

A composite function is a function that is dependent on another function. It is created by putting one function inside another.

General form

We can deduce that “g of x” is a function in terms of variable x for any function, such as g(x).

Domain:

The following are the steps taken to figure out what a composite composition’s domain is (f∘g)(x):

1. Determine the inside function g(x)’s domain .
2. Determine the domain of the external function f(x).
3. Find the x inputs in the domain of the inner function g(x), where g(x) is in the domain of f(x).
4. The resultant set of intersection of both domains will be the composite function’s domain.

In the same way, every form function

“f of g of x” will be read as f(g(x)).

Composition of Function Properties

If there are three functions, f, g, and h, they are said to have been associative if and only if they meet the function composition associative condition:

f∘(g ∘ h) = (f∘g)∘h

• The commutative property states that the two functions f and g are commuted if and only if;

g ∘ f = f ∘ g

A few more properties are:

• A one-to-one functions’ 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).

Creating a function by composition of functions

We can combine functions by performing algebraic operations on them, but we can also build functions by composing functions. We constructed a new function that accepts a day as input and returns a cost as output when we need to calculate the heating cost based on a specific day of the year. 

Composition of functions is the process of merging functions so that the output of one becomes the input of another. A composite function is the product of this process. The following notation is used to denote this combination:

f∘g(x) = f(g(x))

The left-hand side is interpreted as “f composed with g at x,” and the right-hand side is read as “f of g of x.” The two sides of the equation are mathematically equal and have the same meaning. 

The composition operator is represented by the open circle symbol. This operator is generally used to stress the relationship between the functions without mentioning any specific input value. Similar to how addition and multiplication combine two numbers to generate a new one, the composition is a binary operation that combines two functions to create a new one. It is vital to not mix up function composition with multiplication since, as we saw earlier, 

f(g(x)) ≠ f(x)g(x) .

Composition of Functions Examples

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 three categories: 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.

Solution: 

Given, f(x) = 2x

g(x) = x+ 1

As a result, the composition of f derived from g will be:

(f∘g)(x) = f(g(x)) = f(x+1) = 2(x+1)

Using the value of x = 1 as an example

f(g(1)) = 2(1+1) = 2 (2) = 4

3. Given f (x) = 2x + 3, find (f ∘ f) (x).

Solution:

(f ∘ f) (x) = f[f(x)]

= 2(2x + 3) + 3

= 4x + 9

Derivative of Composite Function

Calculus is incomplete without derivatives. They aid in the computation of the rate of change, maxima, and minima for functions. Limits are used to define derivatives, which is known as the initial form of the derivative. 

We already know how to calculate derivatives of composite functions, but there are occasions when we need to deal with sophisticated mathematical functions that are made up of many functions. Calculating the derivative for such functions by brute force becomes difficult and time-consuming. It is critical to understand the rules and methods that facilitate our calculations. One of them is the chain rule, which enables us to compute the derivatives of complex functions. Let’s take a look at this regulation in more detail.

Chain Rule and Composite Functions

Let’s imagine that we wish to determine the derivative for the function f(x) = (x + 1)2. These functions are referred to as composite functions since they are made up of multiple functions. They are usually of the form g(x) = h(f(x)), but they can also be expressed as g = hof (x). The given function f(x) = (x + 1)2 is made up of two functions in our situation.

f(x)=g(h(x)) where g(x)=x2 and h(x)=x + 1.

The Chain Rule

Consider two differentiable functions with a common domain, f(x) and g(x). The derivative of a function created by composing these two functions, f(g(x)), is thus given by:

d f(g(x))dx = f′(g(x)).g′(x)

It essentially requires you to differentiate the outer function, f(x), first before moving on to the inner function. Then, multiply the result by the inner function’s derivative, which is g(x) in our case.

Conclusion

A composite function can be computed by first computing the inner function with the supplied input value and then computing the outer function with the inner function’s output as its input.

• A table can be used to assess a composite function.
• A graph can be used to evaluate a composite function.
• A formula can be used to evaluate a composite function.

The inputs in the domain of the inner function that correlate to the outputs of the inner function in the domain of the outer function make up the domain of a composite function.

Composite functions can be dissected into smaller functions in the same way that functions can be joined to generate composite functions.