CREATE PIEWISE DEFINITION FOR ABSOLUTE VALUE FUNCTION

Create a piecewise definition for the given absolute value function.

Problem 1 :

f(x) = |x + 1|

Solution :

f(x) = |x - 1|

f(x) = x - 1  and  f(x) = -(x - 1)

Case 1 :

f(x) = x - 1

when x ≥ 1, f(x) is positive

Case 2 :

f(x) = -(x - 1)

when x < 1, f(x) is negative

Distributing negative, we get

f(x) = - x + 1

So, the required definition is

Problem 2 :

f(x) = |x - 4|

Solution :

f(x) = |x - 4|

f(x) = x - 4 and  f(x) = -(x - 4)

Case 1 :

f(x) = x - 4

when x ≥ 4, f(x) is positive

Case 2 :

f(x) = -(x - 4)

when x < 4, f(x) is negative

Distributing negative, we get

f(x) = - x + 4

So, the required definition is

Problem 3 :

f(x) = |4 - 5x|

Solution :

f(x) = |4 - 5x|

f(x) = 4 - 5x and  f(x) = -(4 - 5x)

Case 1 :

f(x) = 4 - 5x

when x < 4/5, f(x) is positive

Case 2 :

f(x) = -(4 - 5x)

when x ≥ 4/5, f(x) is negative

Distributing negative, we get

f(x) = - 4 + 5x

So, the required definition is

Problem 4 :

f(x) = |3 - 2x|

Solution :

f(x) = |3 - 2x|

f(x) = 3 - 2x and  f(x) = -(3 - 2x)

Case 1 :

f(x) = 3 - 2x

when x < 3/2, f(x) is positive

Case 2 :

f(x) = -(3 - 2x)

when x ≥ 3/2, f(x) is negative

Distributing negative, we get

f(x) = - 3 + 2x

So, the required definition is