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Composite Function Calculator

Composite Function Calculator

Enter values of functions and points to get the instant composition of functions ((f o g)(x), (f o f)(x), (g o f)(x), and (g o g)(x)) at different points with this tool.

Enter f(x)

Enter g(x)

Enter a point:

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Composite function calculator helps you to solve the composition of the functions from entered values of functions f(x) and g(x) at specific points. Get step by step calculations that help you understand how to compose a reduced function from given complex functions.

How Does Composite Function Calculator Work?

The procedure is very simple. Our composite functions calculator is programmed to give you instant outcomes by providing a couple of values. Let’s dig more!

What You Need To Enter?

  • Please provide the values of function f(x) and g(x)
  • After that, enter the point at which you want to compose a new function

What You Can Get Here!

  • Composite Function: Our f of g calculator will determine the following compositions for the input functions you provide:
    • (f∘g)(x)
    • (g∘f)(x)
    • (f∘f)(x)
    • (f∘g)(x)
  • Detailed Steps: Get a complete solution step by step that will help you understand it better.

What Is Composite Function?

Function composition is a mathematical procedure that allows you to apply the function f(x) through the result of function g(x).

Brief Concept:

Function composition is done by substitution of one function into the other function.

For Instance, g (f(x)) is the composition functions of f (x) and g (x). The composite functions g (f(x)) is pronounced as “f of g of x” or “f compose g”.

Where the function f (x) is used as inner function and g (x) function is called as an outer function.

Moreover, we also read g (f(x)) as “the function f (x) is inner function of the outer function g(x)


Look at the following picture:

  • The result of g(x) is sent through f(x)
  • It is written as (g∘f)(x) which means f(g(x))

Solve the Composition of Functions (Step-by-Step):

To determine the composition of two different functions, we use a circle (o) between the functions for composition. So f o g is pronounced as f compose g, and g o f is as g compose f respectively.

Apart from this, we can plug one function into itself like f o f and g o g. Here are some steps that tell how to do function composition:

  • First write the composition in any form like \( (go f) (x) as g (f(x)) or (g o f) (x^2) as g (f(x^2))\)
  • Put the value of x in the outer function with the inside function then just simplify the function

Put the value of x in the outer function with the inside function then just simplify the function.

Although, you can manually determine composite functions by following these steps to make it convenient for you. The composite function calculator will do all these compositions for you by simply entering the functions.

Domain of Composite Functions:

Composite Function Calculator
The domain is the value that we give to any function to analyze its behavior against it. Our composite function gives you a newly composed function only if the domain of the given functions is the same.

Let us discuss what does it mean!

The domain of composite functions g(f(x)) is always dependent on the domains of the g(x) function and the domain of the f(x) function.

Suppose if you are having (g∘f)(x) = g(f(x)), then remember that:

  • You get the right domain for f(x)
  • Also, make sure you get the right domain for g(x) as well


If f(x) = 1/(x+2) and f(x) = 1/(x+3)

Then what is the domain of the composite function f(g(x))?


The inner function in the f(g(x)) has the following domain:

Domain {g(x)} = {x l x ≠ -3}

So we will solve for f(g(x)):

f(g(x)) = f(1/(x+3))

f(g(x)) = f(1/((1/(x+2))+3))

f(g(x)) = 1/1+2x+6/x+3

f(g(x)) = x+3/2x+7

Therefore, the domain of f(g(x)) is:

Dom {f(g(x))} = {x : x ≠ -7/2}

Range of Composite Functions:

The range of the composite function determined with the function composition calculator does not depend upon the inner and outer functions:


Consider the same function in the above example:

f(g(x)) = x+3/2x+7

(2x + 7) y = x + 3

2xy + 7y = x + 3

2xy – x = 3 – 7y

x (2y – 1) = 3 – 7y

x = (3 – 7y) / (2y – 1)

Range = {y : y ≠ 1/2}


What Is the De-Composing of a Function?

The process of breaking a function into the composition of other functions. For example,
(x+1/x^2)^4 this function made from a composition of two functions are:

f(x) = x + 1/x^2

g(x) = x^4

And we get:

(g o f) (x)= g (f(x)) = g(x + 1/x) = (x + 1/x^2)^4

What Is An Iterated Function?

The function that repeats compositions of a function with itself is called iterated function like

(g ∘ g ∘ g) (x) = g (g (g (x))) = g^3(x)


From the source of Wikipedia: Composite functions, Composition monoids, Functional powers, Composition operator.

From the source of the story of mathematics: Solve Composite Functions, Alternative notations, Multivariate functions.

From the source of PurpleMath: Composing Functions at Points, the domain of a composite function.