Verify formulas for the function f(x) = x^2

by RoRi

July 13, 2015

Establish the following formulas for the function $f(x) = x^2$ defined for all $x \in \mathbb{R}$ .

Proof.
$f(-x) = (-x)^2 = x^2$ and $f(x) = x^2$ ; hence, $f(x) = f(-x)$ , for all $x \in \mathbb{R}. \qquad \blacksquare$
Proof.
$f(y) - f(x) = y^2 - x^2 = (y-x)(y+x)$ for all $x \in \mathbb{R}. \qquad \blacksquare$
Proof.
$f(x+h) - f(x) = (x+h)^2 - x^2 = x^2 + 2xh + h^2 - x^2 = 2xh+h^2$ , valid for all $x \in \mathbb{R}. \qquad \blacksquare$
Proof.
$f(2y) = (2y)^2 = 4y^2 = 4f(y)$ , valid for all $y \in \mathbb{R}. \qquad \blacksquare$
Proof.
$f(t^2) = (t^2)^2 = t^4$ and $f(t)^2 = (t^2)^2 = t^4$ . This is valid for all $t \in \mathbb{R}. \qquad \blacksquare$
Proof.
$\sqrt{f(a)} = \sqrt{a^2} = \pm a = |a|$ valid for all $a \in \mathbb{R}. \qquad \blacksquare$