# Function composition calculator | Create a new function by composition of functions

## Function composition calculator

$$f(x)=$$

$$g(x)=$$

The Function Composition Calculator is an excellent tool to obtain functions composed from two given functions, (fāg)(x) or (gāf)(x).

To perform the composition of functions you only need to perform the following steps:

1. Select the function composition operation you want to perform, being able to choose between (fāg)(x) and (gāf)(x).
2. Enter the functions f(x) and g(x) in their respective fields. You can check below the list of functions that you can use to write the functions.
3. Then you need to press the Calculate button.
The solution explained step by step will automatically be displayed.
4. In the solution, the option of evaluating the resulting composite function for a given value of x is given.

Following is the list of functions that you can use in the function composition calculator:

• sin(x) ā sine
• cos(x) ā cosine
• tan(x) ā tangent
• sec(x) ā secant
• csc(x) ā cosecant
• cot(x) ā cotangent
• asn(x) ā arcsine
• acs(x) ā arcsine
• atn(x) ā arctangent
• asc(x) ā arcsecant
• acs(x) ā arcsecant
• act(x) ā arctangent
• snh(x) ā hyperbolic sine
• csh(x) ā hyperbolic cosine
• tnh(x) ā hyperbolic tangent
• sch(x) ā hyperbolic secant
• cch(x) ā hyperbolic cosecant
• cth(x) ā hyperbolic cotangent
• ash(x) ā hyperbolic arcsine
• ach(x) ā hyperbolic arcsine
• ath(x) ā hyperbolic arctangent
• ln(x) ā natural logarithm
• lne(x) ā natural logarithm
• lgn(n,x) ā logarithm of base n
• log(x) ā natural logarithm
• abs(x) ā absolute value
• sqt(x) ā square root
• sqrt(x) ā square root
• cbt(x) ā cube root

Arithmetic operators:

• + ā add
• - ā subtract/negative sign
• * ā multiply
• / ā divide
• ^ ā exponent

## What is a composite function? | Composite function definition

Function composition, also known by the term composite function, consists of combining two or more functions in such a way that one function becomes the argument of the other. If we have two or more functions that are contained one inside the other, we call them composite functions.

The notation used for function composition is:

(f ā g)(x)=f(g(x))

The value of the composite function at x is equal to the function f evaluated at g(x).

## How to do composite functions

We now proceed to explain how to obtain a composite function from two functions using the following example:

Example 01: Determine the composite function (fāg)(x) if f(x)=x2+1 and g(x)=2x+3.

As we said, the composition (fāg)(x) implies substituting the independent variable of the function f(x) by the function g(x), that is, (fāg)(x)=f(g(x)), therefore, in x2+1 we will replace the variable x with the expression 2x+3. The result is as follows:

• (fāg)(x)=f(g(x))
• (fāg)(x)=(2x+3)2+1
• (fāg)(x)=4x2+12x+9+1
• (fāg)(x)=4x2+12x+10

If we now want to determine the composite function (gāf)(x), what we do is work with 2x+3, substituting the value of x for x2+1:

• (gāf)(x)=g(f(x))
• (gāf)(x)=2(x2+1)+3
• (gāf)(x)=2x2+2+3
• (gāf)(x)=2x2+5

## Composite functions worksheet

Here is a worksheet so you can practice composition of functions. The solutions of each exercise are located on the last page of the document.

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