Composition of Functions

The composition of a function is a process in which two functions, f and g, are combined to produce a new function, h, with the formula h(x) = g(f(x)). It means that the g function is being applied to the x function. In other words, a function is applied to the output of another function.

Let g: A→ B and f: B→ C be two functions with the same range but different domains. Another function defined as fog is the composite of the functions f and g.

y = (fog) = f(g(x))

Representation :

The symbol for the function composition is. It can also be shown without the use of this sign, by using brackets instead. i.e.,

(fog) = f(g(x)) is written as “f of g of x.” The inner function is g, and the outer function is f.

(gof)=g(f(x)) is written as “g of f of x.” The inner function is f, and the outer function is g.

Properties of Composition of Functions :

Associative Property : If k : AB ; g : BC and f : CD then, 

 (fog)ok = fo(gok)

Commutative Property :  If g : AB and f : BC, then

fog gof

Injective Functions : If g : AB and f : BC are injective functions, then

fog is also an injective function

Surjective Function :  If g : AB and f : BC are surjective functions, then

fog is also an surjective function

Understanding with representation :

Let us look at a function

If f(x)=x2+1 and g(x)=cos x then;

g(f(x)) = g(x2+1)

= cos (x2+1)

Derivatives of Composite Functions :

The chain rule approach is used to evaluate derivatives of composite functions (also known as the composite function rule). ‘Let h be a real-valued function that is a composite of two functions f and g,’ says the chain rule. h = f o g, for example. If u = g(x) and du/dx and df/du exist, then this might be written as:

d(h(x))/dx = df/du du/dx = Derivative of h(x) w.r.t. x = Derivative of f(x) w.r.t. u Derivative of u w.r.t. x

Using the chain rule formula, another approach to write the derivatives of composite functions is: w.r.t. x = Derivative of f(x) w.r.t. g(x) Derivative of g(x) w.r.t. x d(f(g(x))/dx = f’ (g(x)) g’ (x). The derivative of a composite function is the product of the derivative of the outside function with regard to the inner function and the derivative of the inside function with respect to the variable, to put it another way.

Examples :

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

(f ∘ f) (x) = f[f(x)]

= 2(2x + 3) + 3

= 4x + 9

2.Find (g ∘ f) (x) given that, f (x) = 2x + 3 and g (x) = –x2 + 5

⟹ (g ∘ f) (x) = g [f (x)]

Replace x in g(x) = –x2 + 5 with 2x + 3

= – (2x + 3)2 + 5

= – (4x2 + 12x + 9) + 5

= –4x2 – 12x – 9 + 5

= –4x2 – 12x – 4

Conclusion :

The following two steps are required to apply the composition fg to an input x. The function g is first applied to the input x, and the result g(x) is returned as the output. Following that, you apply the function f to the input g(x) and get the return f(g(x)). The composition can be written as (fog)(x)=f(g(x)).