# Composition of Functions Calculator



A calculator for the composition of functions is presented.

## Definition of Composition of Functions

In the diagram below, function $f$ has another function $g$ as an input. Starting from the input $x$, applying function $g$ then function $f$, we end up with a function called the composite function or composition of $f$ and $g$ denoted by $f_o g$ and is defined by

$(f_o g)(x) = f(g(x))$

This composite function is defined if $x$ is in the domain of $g$ and $g(x)$ is in the domain of $f$. (see digram below).
According to the definition above, to find the composition $(f_o g)(x)$, we substitute the variable of $f$ by $g(x)$
Example
Let $f(x) = x^3+2x^2 - 3x -1$ and $g(x) = x + 2$. Find the composition $(f_o g)(x)$
Solution
Definition
$(f_o g)(x) = f(g(x))$
Substitute the variable $x$ in $f$ by $g(x)$
$= (g(x))^3+2(g(x))^2 - 3(g(x)) -1$
Substitute $g(x)$ by its formula
$= (x+2)^3+2(x+2)^2 - 3(x+2) -1$
Expand
$(f_o g)(x) = x^3+8x^2+17x+9$

## Use of the Composition Calculator

1 - Enter and edit functions $f(x)$ and $g(x)$ and click "Enter Functions" then check what you have entered and edit if needed.
2 - Press "Calculate Composition".
Note that the five operators used are: + (plus) , - (minus), / (division) , ^ (power) and * (multiplication). (example: f(x) = x^3+2*x^2 - 3*x -1 ).(more notes on editing functions are located below)

$f(x)$ =

$g(x)$ =

Notes: In editing functions, use the following:
1 - The function square root function is written as (sqrt). (example: sqrt(x^2-1)
2 - The exponential function is written as (e^x). (Example: e^(2*x+2) )
3 - The log base e function is written as ln(x). (Example: ln(2*x-2) )
Here are some examples of functions that you may copy and paste to practice:
sqrt(x)       x^2 + 2x - 3       (x^2+2x-1)/(x-1)       1/(x-2)       ln(2*x - 2)
sqrt(x^2-1)       2*sin(2x-2)       e^(2x-3)       1/sqrt(x^2-1)       x/(x+1)

## More References and Links

Composition of Functions
Composition of Functions Questions with Solutions
Algebra and Trigonometry - Swokowsky Cole - 1997 - ISBN: 0-534-95308-5
Algebra and Trigonometry with Analytic Geometry - R.E.Larson , R.P. Hostetler , B.H. Edwards, D.E. Heyd - 1997 - ISBN: 0-669-41723-8