2.7 SOME BASIC FUNCTIONS AND THEIR PROPERTIES

What do we mean when we say that 3.14 is an approximation of π ? This means that on the real line these two numbers are ‘close’ to each other. The numbers −7.1 and −7 are closer to each other than the pair 10 and −1000. The modulus function is useful to measure the distance between two numbers. |x| is just the distance between origin and the point x. We formally define the modulus function || : ℝ → ℝ+ as follows.

2.7.1 The Modulus Function

 

|x| = x, x ≥ 0     (2.11)

 

       = −x, x   < 0.        (2.12)

 

The distance between two points x and y is nothing but |x − y|. The statement |x| ≤ a is equivalent to the statement −a ≤ x ≤ x.

PRACTICE 2.31

Show that the function f : ℝ → ℝ defined as f(x

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