Inverse function : Inverse (function)
In mathematical analysis, an inverse function is in simple terms a function which "does the reverse" of a given function.
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 : x → 3x + 2
- f -1 : x → (x - 2) / 3
The superscript "-1" isn't an exponent. Similarly, f 2(x) means "do f twice", that is f(f(x)), not the square of f(x) (sadly, thisn'tation has an exception for the trigonometric functions: sin2(x) usually does mean the square of sin(x). As such, the prefix arc is sometimes used to denote inverse trigonometric functions[?], eg arcsin x for the inverse of sin x).
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.
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.
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)
- y = f -1(x).
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place. Thirdly, there are the Arab traders, whose business is to go
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darker-skinned than the Egyptians and the Bedouin. In the past many
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the towns are brick walls with gates that are shut at night for fear
are much poorer. They can buy things from the traders in the caravans
make many things to give in exchange, most people have to be content
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