# functions.

• December 8th 2010, 03:43 PM
pedroiscool7
functions.
what are functions?(Bear)(Bear)(Bear)(Bear)
• December 8th 2010, 04:01 PM
dwsmith
Functions are one-to-one and onto mappings from a set A to a set B.

$\displaystyle f:A\rightarrow B$

Functions also have an inverse.

Example $\displaystyle e^x$ is the inverse to $\displaystyle ln(x)$
• December 8th 2010, 06:15 PM
rtblue
Here is a beginner's guide to functions:

Functions versus Relations
• December 9th 2010, 02:44 AM
HallsofIvy
Quote:

Originally Posted by dwsmith
Functions are one-to-one and onto mappings from a set A to a set B.

$\displaystyle f:A\rightarrow B$

Functions also have an inverse.

Example $\displaystyle e^x$ is the inverse to $\displaystyle ln(x)$

This is not true. The standard definition of function does NOT require "one to one" nor "onto". A function, from set A to set B, is a set of ordered pairs, (x, y), with x from A, y from B, such that we do not have two distinct ordered pairs with the same first element- that is, we cannot have (x, y1), (x, y2) with the same x but different y1 and y2. In more common terms that means that a function assigns a value of y, from set B, to every x in set A such that we do NOT assign different values of y to the same x.

Here's what Wikipedia says: http://en.wikipedia.org/wiki/Function_(mathematics)
• December 9th 2010, 01:14 PM
dwsmith
Strictly speaking, a "well-defined" function associates one, and only one, output to any particular input.

If not, then it is a multivalued-function. Multivalued function - Wikipedia, the free encyclopedia

My definition corresponds to, as wikipedia calls it, a well-defined function then.
• December 10th 2010, 03:05 AM
HallsofIvy
Sorry, but, no, it doesn't. Yours corresponds to "invertible function". The function, for example, $f(x)= x^2$, is a "well defined function", with one and only one output to a particular input but is NOT "one to one and onto" because it does not asociate one and only one input with a particular output.