Definition of Signum function

The signum function simply gives the sign for the given values of x. For x value greater than zero, the value of the output is +1, for x value lesser than zero, the value of the output is -1, and for x value equal to zero, the output is equal to zero. The signum function can be defined and understood from the below expression.

Properties of signum function

Let us consider x. The function sgn x yielding a real number, is defined by:
sgn x = 1 if 0 < x, sgn x = -1 if x < 0, sgn x = 0, otherwise.

The following propositions are true:

Solved Examples:

Example 1: Find the result for the values of x, using the signum function

x = {- 4.93, - 7.66, 12, 0, 4.2, 2.33333, -8.10}

Solution: Here we use the following signum function to find the output, for the input values of x.

x = {- 4.93, - 7.66, 12, 0, 4.2, 2.33333, -8.10}

Output = {-1,-1,+1,0,+1,+1,-1}

Now, -1 ≤ sinx ≤ 1

⟹ 0 ≤ sinx + 1 ≤ 2

⟹ 0 ≤ (sinx + 1)² ≤ 4

⟹ 2 ≤ (sinx + 1)² + 2 ≤ 6

Let’s take a some negative values as the domain of this function x = {-2,- 3,- 5}.

Applying these values for the signum function we have f(-2)= -1,

f(-3)= -1,f(-5)= -1.

Here, the range or the answer for all the negative values of x is always -1, which is a constant value.

Hence proved.

Solution:
To find the domain let us look at every values x can take here.
We found that division by zero isn't defined.
So the denominator of a function can't be zero.
This gives, x-3≠0
Addition of 3 on both sides, we get, x≠3
So the main value which couldn't be taken by x is 3.
This gives the domain contains the set of all real numbers aside from 3, that is R-{3}..
Now the range is the set of all values taken by f(x).