Science Library - free educational site

Operations with Functions

Composition of functions

Let $f$ and $g$ be two functions in $x$, where $x$ ∈ ℝ, then:

1. $(f±g)(x) = f(x) ± g(x)$

2. $a(f(x)) = af(x)$, $a$ ∈ ℝ

3. $(fg)(x) = f(x)g(x)$

4. $(f/g)(x) = {f(x)}/{g(x)}$, $g(x)$ ≠ 0

The composition of two functions, g and h, such that g is applied first and h second, is given by:

(g o h)(x) = g(h(x))

The domain of the composite function (g o h) is the set of all x in the domain of h such that h(x) is in the domain of g.


e.g. Let f(x) = √x, and g(x) = $x^2$.

(f o g)(x) = f(g(x)) = $√{x^2} = x$

(g o f)(x) = g(f(x)) = $(√x)^2 = x$

e.g. Let $f(x) = √{x+4}$, and g(x) = $x^2$.

(f o g)(x) = f(g(x)) = $√{(x^2+4)} ≠ x + 4$

(g o f)(x) = g(f(x)) = $(√{x+4})^2 = x + 4$

From these examples, we can see that (f o g)(x) is not always equal to (g o f)(x)!

Decomposing Composite Functions

The function that is applied first is the 'inside' function, and that applied second the 'outside' function. e.g. in $f(x) = (x+3)^2 = (g(h(x))$, the inside function is $h(x) = x+3$, and the outside function is $g(x) = x^2$.

Domains of Composite Functions
Composite functions have restricted interdependent domains

If x is in the domain of the composite function (g o h), then x must be in the domain of h, and h(x) must be in the domain of g.

The inverse of a function $f(x)$ is written as $f^{-1}$. The effect of the inverse of a function is to reverse the action of the function: if x undergoes a transformation to y through function $f(x)$, then applying $f^{-1}(x)$ to y will return the original value of x.

For example, if $f(x) = 2x + 4$, then $f^{-1}(x) = {x - 4}/2$.

Not all functions have an inverse. Be careful, $f^{-1}(x)$ is not the same as $[f(x)]^{-1}$. $f^{-1}(x)$ is a reflection across the line $y = x$, and $[f(x)]^{-1}$ is a reflection across the x-axis.

Functions that do not have an inverse can be tested by the horizontal line test.

If any horizontal line crosses the graph of a function more than once, then the function has no inverse. This is because the inverse is not a function, but a relation (2 values of y for one value of x.

An example of a function that has no inverse is $f(x) = x^2$, while $f(x) = x^3$ does have an inverse.

Inverse function
An inverse function is a reflection across the line y = x

Finding inverses algebraically

If $f(x) = 4x - 6$, the inverse can be found by setting y as $f(x)$, then exchanging the places of x and y:

$y = 4x - 6$, so $x = 4y - 6$ for the inverse.

Rearranging to return y to the left:

$y = {x + 6}/4$

This can be written as the inverse: $f^{-1}(x) = {x + 6}/4$

To check that a function is the inverse of another function, combine the functions:

$$(f o f^{-1})(x) = x$$

Example above: $(f o f^{-1})(x) = 4({x + 6}/4) - 6 = x$

A self-inverse function is a function whose inverse is the same. e.g. $f(x) = a/x$ and $f(x) = a - x$, where a ∈ ℝ.

Identity Function

$$(f o f^{-1}) = I$$

where $I(x) = x$. The identity function leaves $x$ unchanged.

Content © Andrew Bone. All rights reserved. Created : September 26, 2014 Last updated :July 20, 2015

Latest Item on Science Library:

The most recent article is:

Air Resistance and Terminal Velocity

View this item in the topic:


and many more articles in the subject:

Subject of the Week


Environmental Science is the most important of all sciences. As the world enters a phase of climate change, unprecedented biodiversity loss, pollution and human population growth, the management of our environment is vital for our futures. Learn about Environmental Science on

Environmental Science

Great Scientists

Francis Crick

1916 - 2004

Francis Crick was a physicist who worked with molecular biologist James Watson to discover the structure of the DNA molecule in 1953. Their work heralded the 'coming of age' of the biological sciences, and permitted the later breaking of the 'code of life'.

Francis Crick

Quote of the day...

ZumGuy Internet Promotions