The function defined as:

is called an absolute value function.

Note :      √x² = |x| ∀  x  ε R

        The graph of an absolute value function is shown in the figure given above. Its properties are:

(i)     An absolute value function is an even function

(ii)    It is strictly increasing in [0, ∞) and strictly decreasing in (-∞, 0].

Illustration 12: Draw the graph of the following functions.

(a)    y = |x - 1| + |x - 4|

(b)    y = |sin x|

(c)    y = sin |x|

(a)    Note:  x - 1 = 0 => x = 1 and x - 4 = 0 => x = 4 i.e. y changes its definition at x =1 and x = 4.

        y = |x - 1| + |x - 4|

        let - ∞ < x < 1

        y = -(x - 1) - (x - 4) = -2x + 5

Now, let 1 < x < 4

        y = (x - 1) - (x - 4) = 3

Again,        Let 4 < x

        y = (x - 1) + (x - 4) = 2x - 5

                         


(b)    y = |sin x|

        y > 0 ∀ x ε R

(c)    y = sin |x|

        A x > 0,                     y = sin x

        A x < 0,                     y = sin (-x) = -sinx