Constant Function: Let k be a fixed real number. Then a function f (x) given by f (x) = k for all x ϵ R is called a constant function. Sometimes we also call it the constant function k.
Identity Function: The function defined by I (x) = x for all x ϵ R, is called the identity function on R.
Modulus Function: The function defined by f (x) = |x| \(=\left\{ \begin{align}&x,\,\,\,when\,\,\,x\ge 0 \\& -x,\,\,when\,\,\,x<0 \\\end{align} \right.\) is called the modulus function.
Properties of Modulus Function: The modulus function has the following properties:
i) For any real number x we have √x2 = |x|.
ii) If a, b are positive real numbers then
⇒ x2 ≤ a2 |x| ≤ a ⇔ -a ≤ x ≤ a
⇒ x2 ≥ a2 ⇔|x| ≥ a ⇔ x ≤ -a or x ≥ a
⇒ x2 > a2 ⇔|x| > a ⇔ x < -a or x > a
⇒ a² ≤ x² ≤ b² ⇔ a ≤ |x| ≤ b ⇔ x ϵ [-b, -a] U [a, b]
⇒ a2 ≤ x2 ≤ b2 ⇔ a ≤ |x| ≤ b ⇔ x ϵ [-b, -a] U [a, b]
iii) |x + y| = |x| + |y|
⇒ (x ≥ 0 and y ≥ 0) or (x < 0 and y < 0)
iv) |x – y| = |x| – |y|
⇒ (x ≥ 0) and |x| ≥ |y| or (x ≤ 0, y ≤ 0 and |x| ≥ |y|)
v) |x ± y| < |x| + |y|
vi) |x ± y| ≥ ||x| – |y||