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```Absolute Value Function

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|

∀ x > 0,    y = sin x

∀ x < 0,   y = sin (-x) = -sinx

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