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The modulus value of a number is measured to be the distance of the number from its origin. Suppose the distance from point 2 to origin is 2, then the distance from point -2 to origin is 2 as well. Modulus function is also called an absolute value function because it gives the magnitude of a number regardless of the sign. The modulus of a number x is represented by |x|. It gives a non-negative value. In this article, we look more into modulus function, its properties, its graphical representation, and solved problems to get a clear understanding.
Modulus function
A relation is said to be a function if each element of a set has only one image or rage to another set. The modulus function is,
f(x) = |x| or y = |x|.
Where f: R → R and x ∈ R.
If any positive number is given to the modulus function, it remains the same. If any negative integer is given to the modulus function, it changes its sign to positive.
As per this function,
If x = 21 then f(x) = 21, because x is greater than 0
If x = 0, then f(x) = 0, because x is equal to 0
If x = -3, then f(x) = – (-3) = 3, because x is less than 0
Modulus function formula
As we know the modulus value is always positive, if f(x) is a modulus function then,
- f(x) = x, if x is greater than zero
- f(x) = 0, then x is zero
- f(x) = -x if x is less than zero
If the value of x is greater than or equal to zero then the value of x remains the same. But if x is less than zero then the function is multiplied with a negative sign, to get the final product as positive.
Domain and Range of Modulus function
The modulus function can be applied to any real number, and the final output is either zero or positive. Hence the domain of the modulus function lies in between (-∞, ∞), it can take up any number. The range of modulus function is always positive, regardless of the input. Hence, the range of modulus function can be given as R, which is (0, ∞).
Graph of modulus function
The modulus function is given as y = |x|. It can be further briefed as,
y = x, if the value of x ≥ 0.
y = -x, if value of x ≤ 0.
The line equation y – x = 0 is drawn passing through, origin, and points which have equal x and y. The line equation x + y = 0 is drawn through the origin and points in the first quadrant whose magnitudes are equal.
| Value of x | f(x) = |x| |
| -3 | 3 |
| -2 | 2 |
| -1 | 1 |
| 0 | 0 |
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |
Properties of Modulus function
The modulus function always produces a non-negative integer as an outcome. It is a wrong notation to equate modulus function to a negative number.
- |-x| = x
- |x + y| ≤ |x| + |y|
- |x – y| = 0, then x = y
- |x – y| ≥ ||x| – |y||
- |x y| = |x| |y|
- |x/y| = |x| / |y|, if y is not equal to zero.
Examples
- Apply modulus function |x| for x = -3.3 and x = 4.
x = – 3.3, then applying modulus |x| = |- 3.3| = 3.3
x = 4, then applying modulus |x| = |4| = 4.
- Solve |x – 4| = 9 using modulus function.
The following equation is can be evaluated to two different equations,
If x – 4 > 0, then |x – 4| = x – 4.
If x – 4 < 0, then |x – 4| = – (x – 4) = 4 – x.
In case 1, when x – 4 > 0,
|x – 4| = x – 4.
x – 4 = 9
x = 13
In case 2, when x – 4 < 0
|x – 4| = – (x – 4) = 4 – x.
4 – x = 9
x = -5
Hence the values of x are 13 and -5.
Conclusion
Modulus function is pretty much used to convert any given non-negative numbers into positive. It gives the absolute value of a number. The outcome is always positive, regardless of the input. This concept is most popularly used in complex numbers. Proper learning about the properties of modulus function is important as it is helpful while solving problems and boosting them up.
