Function square

The function square is the function which with a Real number X associates its square, noted X ² , that is to say X multiplied by him even . It introduces the functions power, it is one of simplest of them.

Properties

; Sign

the first property is the positivity of the function. Indeed whatever the X real, $y=x\; \backslash \; times\; x$ one has the same sign inevitably twice on the right; thus is higher or equal there 0. And cancels itself only in 0.

; Parité

Vient then the parity from the function i.e. $f\; (X)\; =f\; (-\; X)$. Indeed with the preceding remark $(-\; X)\; \backslash \; times\; (-\; X)\; =x\; \backslash \; times\; x$.

Solution of equation of the type X ² has

See also: quadratic equation

When $x^2\; =\; a$, there are three possible cases:

• : No solution in the whole of realities R
• $a\; =\; 0$: A solution, X = 0
• $a\; >\; 0$: Two solutions, $x\; =\; \backslash \; sqrt\; \{has\}$ or $x\; =\; -\; \backslash \; sqrt\; \{has\}$

For example, if $x^2\; =\; 9$ then $x\; =\; 3$ or $x\; =\; -3$, because $3^2=\; (-\; 3)\; ^2=9$.

Derived

The Dérivée from the function square is $2x$, it is a Fonction closely connected odd.

Chart

In a orthonormal Reference mark, the function is represented by a Parabole whose top is the item (0,0). It is noticed well that the entirety of the parabola is above the curve and the parity is detectable thanks to the axis of symmetry which is the y-axis.

The function square being an even function, its chart admits an axis of symmetry which is the y-axis.

The function square has for limit more the infinite one in more the infinite one and less the infinite one.

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