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

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

  1. f(-x) = f(x).
  2. f(y) - f(x) = (y-x)(y+x).
  3. f(x+h) - f(x) = 2xh+h^2.
  4. f(2y) = 4f(y).
  5. f(t^2) = f(t)^2.
  6. \sqrt{f(a)} = |a|.

  1. 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
  2. Proof.
    f(y) - f(x) = y^2 - x^2 = (y-x)(y+x) for all x \in \mathbb{R}. \qquad \blacksquare
  3. 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
  4. Proof.
    f(2y) = (2y)^2 = 4y^2 = 4f(y), valid for all y \in \mathbb{R}. \qquad \blacksquare
  5. 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
  6. Proof.
    \sqrt{f(a)} = \sqrt{a^2} = \pm a = |a| valid for all a \in \mathbb{R}. \qquad \blacksquare

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