Set of definitions and graphical representations

What is the domain?

Not all values of x are necessarily permitted in a function. The domain, denoted by Df, is the set of real numbers for which f(x) can be calculated.

Two common pitfalls:

• Division by zero is not permitted: for f(x) = 1/x, we cannot take x = 0, so Df = R excluding {0}.
• Square root: for f(x) = √x, x must be ≥ 0, so Df = [0; +∞[.

The representative curve

In a coordinate system, the representative curve Cf of f is the set of points with co-ordinates (x; f(x)) for x ranging over Df. Each point on the curve corresponds to a pair (antecedent; image).

Reading a graph

• To find the image of a number a: start at a on the x-axis, move up (or down) to the curve, then read the y-coordinate.
• To find the antecedent of a number b: start at b on the y-axis, move along the curve, then read the x-coordinate (there may be 0, 1 or several).

Numerical example

For f(x) = x² - 1:

x	-2	-1	0	1	2
f(x)	3	0	-1	0	3

We can see that 0 has two antecedents (-1 and 1), but that -1 has only one antecedent (0).

Pitfall to avoid

A curve represents a function only if any vertical line intersects it at at most one point (otherwise, the same x would have two images, which is prohibited by definition).