# 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).

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