Absolute value Function Definition

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Definition: -By absolute value function of a real number we mean its numerical value.

For example: - Suppose a function is f (a) = | 4 a |

When a = 1 then      f (1 ) = | 4 (1) | = 4

When a = - 1 then    f (- 1) = | 4 (- 1) | = | - 4 | = 4

 
Other example: - If f (x) = | x | + 2, find the value of the function at x = 5 and x = - 5.

Solution: - f ( x) = | x | + 2

When x = 5 then      f ( 5 ) = | 5 | + 2 = 5 + 2 = 7

When x = - 5 then    f ( - 5 ) = | - 5 | + 2 = 5 + 2 = 7

 
Properties of absolute value function: -

If x and y are two real numbers then

i)             | x | = | - x |

For example, if x = 2 then | x | = | 2 | = 2 and | - x | | - 2 | = 2.

Hence | x | = | - x |


 
ii)            | x – y | = | y – x |


iii)           | x – y | = x – y    when x > y

| x – y | = y – x    when x < y




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