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.

Example

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

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

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 $g(x) = x^2$.

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.

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.

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

Content © Renewable-Media.com. All rights reserved. Created : September 26, 2014 Last updated :July 20, 2015

The most recent article is:

View this item in the topic:

and many more articles in the subject:

Mathematics is the most important tool of science. The quest to understand the world and the universe using mathematics is as old as civilisation, and has led to the science and technology of today. Learn about the techniques and history of mathematics on ScienceLibrary.info.

1949 -

Sir Paul Nurse is a British biologist and geneticist, who won a Nobel Prize for the discovery of controlling proteins in cell cycles. He is the current President of the Royal Society and Chief Executive and Director of the Francis Crick Institute.

Website © renewable-media.com | Designed by: Andrew Bone