SOME STANDARD REAL FUNCTIONS:

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 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 ó|x| ≤ a ó -a ≤ x ≤ a

=> x² ≥ a² ó|x| ≥a ó x≤ -a or x ≥a

=> x² > a² ó|x| > a ó x < -a or x > a

=> a2 ≤ x2 ≤ b2 óa ≤ |x| ≤ b óx ϵ [-b,-a] U [a, b]

=> a² ≤ x² ≤ b² ó a ≤ |x| ≤ b ó 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||