Definition, graph and graphical representation of the square function

Definition and calculation of images

What is the square function?

The square function is the function f defined on R (the set of real numbers) by f(x) = x². It maps each real number x to its square, that is, the product of x and itself.

Calculating an image

To calculate the image of a number under f, we substitute x with that number and carry out the calculation.

Examples:

f(3) = 3² = 9
f(-3) = (-3)² = 9
f(0) = 0
f(1/2) = (1/2)^2 = 1/4

A common pitfall

Be careful with the sign: (-3)^2 = (-3) × (-3) = 9, not -9. The square of a negative number is always positive or zero. Do not confuse -3² (which means -(3²) = -9) with (-3)² = 9. The brackets are essential!

Antecedents

Finding a root of a via f amounts to solving the equation x² = a.

If a > 0, there are two roots: √a and -√a.
If a = 0, there is only one root: 0.
If a < 0, there are no antecedents (a square cannot be negative).

Example: the antecedents of 16 are 4 and -4, because 4² = 16 and (-4)² = 16.

Summary table