The Modulus Function

Understanding the Modulus Function

• The modulus function, often denoted as

  x

  , is a real function which essentially provides the ‘distance’ of a number on the number line from the origin.

• A key characteristic of the modulus function is that it always delivers non-negative output values. For any real number x,

  x

  ≥ 0.

• The modulus function is defined as: for any real number x, if x ≥ 0, then

  x

  = x and if x < 0, then

  x

  = -x.

• One way to visualise the modulus function is as a ‘flipping’ of negative values on the number line to their positive counterparts.

Graphing the Modulus Function

• The graph of the modulus function

  x

  is a V-shape, with its vertex at the origin (0,0).

• For positive x, the graph of

  x

  is the same as the graph of y = x. For negative x, the graph of

  x

  is the reflection of the graph of y = x in the y-axis.

• The modulus function is not differentiable at x = 0, because of the sharp corner in the graph at this point.

Properties of the Modulus Function

• The modulus function has the property that

  x + y

  ≤

  x

  +

  y

  for any two real numbers x and y. This is known as the triangle inequality.

• Another key property of the modulus function is symmetry:

  x

  =

  -x

  for any real number x.

Solving Equations with the Modulus Function

• Solving equations involving the modulus function often involves considering two separate cases: one when the argument of the modulus function is non-negative, and one when it is negative.

• For example, to solve the equation

  x

  = a where a > 0, consider the two cases x = a and x = -a.

• To solve inequalities involving the modulus function, it can be helpful to consider the distance interpretation of the function.

Modulus Function in Absolute Value Inequalities

• Using the modulus function is an effective way to solve absolute value inequalities. For example, the inequality

  x - 3

  < 2 means ‘the distance between x and 3 is less than 2’, which can be solved by referring to distances on the number line.