inverse definition

<mathematics> Given a function, f : D -> C, a function g : C -> D is called a left inverse for f if for all d in D, g (f d) = d and a right inverse if, for all c in C, f (g c) = c and an inverse if both conditions hold. Only an injection has a left inverse, only a surjection has a right inverse and only a bijection has inverses. The inverse of f is often written as f with a -1 superscript.

1. Opposite in order, relation, or effect; reversed; inverted; reciprocal; opposed to direct.

2. <botany> Inverted; having a position or mode of attachment the reverse of that which is usual.

3. <mathematics> Opposite in nature and effect; said with reference to any two operations, which, when both are performed in succession upon any quantity, reproduce that quantity; as, multiplication is the inverse operation to division. The symbol of an inverse operation is the symbol of the direct operation with -1 as an index. Thus sin -1 x means the arc whose sine is x.

<geometry> Inverse figures, two points lying on a line drawn from the center of a fixed circle or sphere, and so related that the product of their distances from the center of the circle or sphere is equal to the square of the radius.

<mathematics> Inverse, or Reciprocal, ratio, the ratio of the reciprocals of two quantities. Inverse, or Reciprocal, proportion, an equality between a direct ratio and a reciprocal ratio; thus, 4: 2:: 1/3: 1/6, or 4: 2:: 3: 6, inversely.

Origin: L. Inversus, p. P. Of invertere: cf. F. Inverse. See Invert.