# All Science Fair Projects

## Science Fair Project Encyclopedia for Schools!

 Search    Browse    Forum  Coach    Links    Editor    Help    Tell-a-Friend    Encyclopedia    Dictionary

# Science Fair Project Encyclopedia

For information on any area of science that interests you,
enter a keyword (eg. scientific method, molecule, cloud, carbohydrate etc.).
Or else, you can start by choosing any of the categories below.

# Inverse function

(Redirected from Inverse map)

In mathematics, an inverse function is in simple terms a function which "does the reverse" of a given function. More formally, if f is a function with domain X, then f -1 is its inverse function if and only if for every $x \in X$ we have:

f - 1(f(x)) = f(f - 1(x)) = x.

For example, if the function x → 3x + 2 is given, then its inverse function is x → (x - 2) / 3. This is usually written as:

$f\colon x\to 3x+2$
$f^{-1}\colon x\to(x-2)/3$

The superscript "-1" is not an exponent. Similarly, as long as we are not in trigonometry, f 2(x) means "do f twice", that is f(f(x)), not the square of f(x). For example, if : f : x → 3x + 2, then f 2 : x = 3*((3x + 2)) + 2, or 9x + 8. However, in trigonometry, for historical reasons, sin2(x) usually does mean the square of sin(x). As such, the prefix arc is sometimes used to denote inverse trigonometric functions, e.g. arcsin x for the inverse of sin(x).

### Simplifying rule

Generally, if f(x) is any function, and g is its inverse, then g(f(x)) = x and f(g(x)) = x. In other words, an inverse function undoes what the original function does. In the above example, we can prove f -1 is the inverse by substituting (x - 2) / 3 into f, so

3(x - 2) / 3 + 2 = x.

Similarly this can be shown for substituting f into f -1.

Indeed, an alternative definition of an inverse function g of f is to require that g o f resp. f o g be the identity function on the domain resp. codomain of f.

### Existence

For a function f to have a valid inverse, it must be a bijection, that is:

• each element in the codomain must be "hit" by f: otherwise there would be no way of defining the inverse of f for some elements
• each element in the codomain must be "hit" by f only once: otherwise the inverse function would have to send that element back to more than one value.

If f is a real-valued function, then for f to have a valid inverse, it must pass the horizontal line test, that is a horizontal line y = k placed on the graph of f must pass through f exactly once for all real k.

It is possible to work around this condition, by redefining f's codomain to be precisely its range, and by admitting a multi-valued function as an inverse.

If one represents the function f graphically in an x-y coordinate system, then the graph of f -1 is the reflection of the graph of f across the line y = x.

Algebraically, one computes the inverse function of f by solving the equation

y = f(x)

for x, and then exchanging y and x to get

y = f - 1(x)

This is not always easy; if the function f(x) is analytic, the Lagrange inversion theorem may be used.

The symbol f -1 is also used for the (set valued) function associating to an element or a subset of the codomain, the inverse image of this subset (or element, seen as a singleton).

## See also

03-10-2013 05:06:04
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details
Science kits, science lessons, science toys, maths toys, hobby kits, science games and books - these are some of many products that can help give your kid an edge in their science fair projects, and develop a tremendous interest in the study of science. When shopping for a science kit or other supplies, make sure that you carefully review the features and quality of the products. Compare prices by going to several online stores. Read product reviews online or refer to magazines.

Start by looking for your science kit review or science toy review. Compare prices but remember, Price \$ is not everything. Quality does matter.
 Science Fair Coach What do science fair judges look out for? ScienceHound Science Fair Projects for students of all ages
 All Science Fair Projects.com Site All Science Fair Projects Homepage Search | Browse | Links | From-our-Editor | Books | Help | Contact | Privacy | Disclaimer | Copyright Notice