Solved Problems on Signum Function

Problem 1: Find the probable values of x if sgn(x² – 5x + 6) = 1.

Answer:

We know that the signum function gives a value of 1 when the value supplied to it is greater than 0.

Thus x² – 5x + 6 > 0

(x-3)(x-2) > 0

It is only possible if both (x-2) and (x-3) are either positive or negative.

Case 1: If both are positive

x can assume the values greater than 3 i.e. x ∈ (3, ∞)

Case 2: If both are negative

x can assume the values lesser than 2 i.e. x ∈ (-∞, 2)

Thus the probable values of x are (-∞, 2) U (3, ∞).

Problem 2: Find the probable values of x if sgn(2x² – 7x + 6) = 1.

Answer:

We know that the signum function gives a value of 1 when the value supplied to it is greater than 0.

Thus 2x² – 7x + 6 > 0

(2x-3)(x-2) > 0

It is only possible if both (x-2) and (2x-3) are either positive or negative.

Case 1: If both are positive

x can assume the values greater than 2 i.e. x ∈ (2, ∞)

Case 2: If both are negative

x can assume the values lesser than 3/2 i.e. x ∈ (-∞, 3/2)

Thus the probable values of x are (-∞, 3/2) U (2, ∞).

Problem 3: Calculate the value of the signum function for the values in the set [1, -0.5, 0.435, 0].

Answer:

We know that [Tex]sgn(x) = \begin{Bmatrix} 1, if~ x>0,\\ -1, if~x<0,\\ 0, if~x=0 \end{Bmatrix}  [/Tex]

Thus sgn(1) = 1,sgn(-0.5) = -1,

sgn(0.435) = 1, and

sgn(0) = 0

Problem 4: Calculate the value of sgn(1/x) if sgn(x) = -1.

Answer:

As, sgn(x)sgn(1/x) = 1 if x is not equal to zero.

Thus sgn(1/x)(-1) = 1

Therefore, sgn(1/x) = -1

Signum Function is an important function in mathematics that helps us to know the sign of a real number. It is usually expressed as a function of a variable and denoted either by f(x) or by sgn(x). It may also be written as a sign(x). Signum Function also has applications in various fields such as physics, electronics, and AI due to which it becomes much more important to study signum function.

A signum function is neither a one-one nor an onto function as various elements has the same image and a pre-image has various images in the co-domain and domain set respectively. In this article, we shall discuss the signum function in detail.