An Introduction to the Inverse Function
Definition and representative curve
What is the inverse function?
The inverse function is the function f defined on R excluding 0 (denoted R* or R{0}) by:
f(x) = 1/x
It maps every non-zero real number to its reciprocal. For example:
- f(2) = 1/2 = 0.5
- f(-4) = 1/(-4) = -0.25
- f(0.1) = 1/0.1 = 10
Why is x = 0 excluded?
Division by 0 does not exist. The domain of definition of f is therefore Df = ]–∞; 0[ U ]0; +∞[.
Table of values
| x | -4 | -1 | -0.5 | 0.5 | 1 | 4 |
|---|---|---|---|---|---|---|
| f(x) | -0.25 | -1 | -2 | 2 | 1 | 0.25 |
The curve: a hyperbola
The graph of f is called a hyperbola. It has two branches:
- one for x > 0, or f(x) > 0 (top right)
- one for x < 0, or f(x) < 0 (bottom left)
The curve gets closer and closer to the axes without ever touching them: the x-axis (y = 0) and the y-axis (x = 0) are asymptotes of the curve.
Common pitfall
Do not confuse 1/x (the inverse function) with the square function x² or with -x. Also note: 1/x is never equal to 0, so the curve never intersects the x-axis, even very far from the origin.