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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