The inverse function

Definition

The inverse function is defined by f(x) = 1/x, for all x not equal to 0 (the domain is R excluding 0, denoted by R - {0}).

Variations

The inverse function is decreasing on ]∞; 0[ and is also decreasing on ]0; +∞[ — but note that it is not decreasing on the union of the two intervals.

x	-2	-1	-0.5	0.5	1	2
1/x	-0.5	-1	-2	2	1	0.5

Representative graph

The graph of f(x) = 1/x is a hyperbola consisting of two branches, symmetric about the origin (0, 0): f(-x) = -f(x) for all non-zero x; the function is said to be odd. The axes of the co-ordinate system (x = 0 and y = 0) are asymptotes: the curve approaches them without ever touching them.

Example

For x = 4, f(4) = 1/4 = 0.25. The larger x becomes, the closer 1/x gets to 0 without ever reaching it.

Common pitfall

0 has no image under the inverse function: 1/0 does not exist. Always check that the denominator is not zero before calculating an image or solving an equation containing 1/x.