Modulus

Modulus Functions

• Modulus function, represented as

  x

  , is the distance of x from the origin on the number line, whether x is positive or negative.

• The graph of a modulus function

  f(x)

  consists of the graph of y = f(x) for f(x) ≥ 0 and the reflection in the x-axis for f(x) < 0.

• The modulus of a sum is not equal to the sum of the moduli:

  a + b

  ≠

  a

  +

  b

  .

• The modulus of a product equals the product of the moduli:

  ab

  =

  a

  *

  b

  .

• For all real numbers a and b, when b > 0, inequalities involving the modulus can be written as -b ≤ a ≤ b.

• Solving modulus equations involves considering both the positive and negative possible values for the expressions within the modulus.

• When graphing modulus functions, remember that the graph is mirrored along the x-axis for negative y-values.

• For complex numbers, modulus is calculated as the square root of the sum of the squares of its real and imaginary part.

• The modulus of a complex number z = a + bi is denoted by

  z

  and is given by √(a² + b²).

• Absolute value inequalities involving the modulus can be written in two ways:

  a

  < b is equivalent to -b < a < b , and

  a

  > b is equivalent to a < -b or a > b.