The modulus function generally refers to the function that gives the positive value of any variable or a number. Also known as the absolute value function, it can generate a non-negative value for any independent variable, irrespective of it being positive or negative. Commonly represented as : y = |x|, where x represents a real number, and y = f(x), representing all positive real numbers, including 0, and f:R→R and x ∈ R

The expression in which a modulus can be defined is:

f(x) = \[\left\{\begin{matrix} x & if x \geq 0\\ -x & if x < 0 \end{matrix}\right.\]

Here, x represents any non-negative number, and the function generates a positive equivalent of x. For a negative number, x<0, the function generates (-x) where

-(-x) = positive value of x.

However, there are different cases for a modular function and can mean differently for various contexts.

Case 1:

For y = |x|, where x is a real number, i.e. x > 0, since variables can have real values only. Here the modulus function of the real variable stays the positive value of the real number.

For x = 2,

y = |2| i.e. = 2.

Case 2:

For y = |f(x)|, here we use f(x) instead of |x|, and therefore the modulus changes the function value and properties, modifying the overall function. A few examples are:

|f(x)| = a ; a > 0 => f(x) = \[\pm\] a

|f(x)| = a; a = 0 => f(x) = 0

|f(x)|= a; a < 0 => There is no solution of this equation

In modulus function, every time |x| = 4, the value of x = ±4

For plotting the graph, we need to take certain values first,

When x = -5 then y = |-5| = 5

Similarly, for x = -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, the respective values of y will be = -4, -3, -2, -1, 0, 1, 2, 3, 4, 5

(Image to be added soon)

Here for x > 0, the graph represents a line where y = x. Similarly for x < 0, the graph is a line where y = -x.

Here both the graph lines hold true the definition of modulus functions. The graph defines the domain and range of modulus function, i.e.

the domain = R (or Real Numbers)

Range = [0,∞] ; where the range of modulus function is the upper half of the Real numbers (R+), including 0.

As the modulus function is understood as a non-negative value, therefore it can be said that the modulus of a variable is similar to that of the square root of the square of the variable. Here's how:

|x| = \[\sqrt{x^{2}}\]

Here are some other non-negative expressions that can explain the non-negative value of the modulus function:

The even exponent of an expression or variable can be defined as: x2n , where n ∈ Z

The even root of a variable can be defined as: x1/2n , where n ∈ Z

The value of y can be defined as: y= 1-sinx, or, y=1-cosx, (since sinx ≤ 1 and cosx ≤1)

Since the modulus function can be effective to find inequality between the numbers, here are the following properties of modulus function:

When a>0,

Here, x lies between -a and a, not considering the end points of the interval, i.e.

|x| < a; a >0 ⇒ -a < x < a

|x| > a; a >0 x < - a or x > a ⇒ x ∈ (- ∞, a) ∪ (a, ∞)

Since the inequalities can be useful to express intervals in the compact form, here's an example of the cosec trigonometric function range that is defined as x ∈ (−∞, -1] ∪ [1, ∞}, represented as:

|x| ≥ 1

|f(x)| < a; a > 0, ⇒ -a < f(x) < a

For x, y as real variables:

|x-y| = 0, ⇔ x=y.

For x, y as real variables, then:

|x+y| ≤ |x| + |y|

|x-y| ≥ ||x| - |y||

For x, y as real variables, then:

|xy| = |x| * |y|

|x/y| = |x| / |y|, where |y| ≠ 0.

For p and q as positive real numbers:

x2 ≤ p2 ⇔ |x| ≤ p ⇔ -p ≤ x ≤ p

x2 ≥ p2 ⇔ |x| ≥ p ⇔ x ≤ -p , x ≥ p

x2 < p2 ⇔ |x| < p ⇔ -p < x < p

x2 > p2 ⇔ |x| > p ⇔ x < -p, x > p

p2 ≤ x2 ≤ q2 ⇔ p ≤ |x| ≤ q ⇔ x ∈ [-q,-p] ∪ [p,q]

p2 < x2 <q2 ⇔ p < |x| < q ⇔ x ∈ (-q,-p) ∪ (p,q)

Example 1: A function f is defined on R as:

f(x) = \[\left\{\begin{matrix} \frac{|x|}{x}, & x\neq 0\\ 0, & x = 0 \end{matrix}\right.\]

Plot the graph

Solution:

When x is a positive integer, the function can be defined as:

f (x) = \[\frac{|x|}{x}\] = \[\frac{x}{x}\] = 1

When x is a negative integer, the function can be defined as:

f(x) = \[\frac{|x|}{x}\] = \[\frac{-x}{x}\] = -1

Therefore, the f can be redefined as:

f(x) = \[\left\{\begin{matrix} 1, & &x>0 \\ 0 & & x=0\\ -1& & x<0 \end{matrix}\right.\]

(Image to be added soon)

The filled dot at (0,0), and the hollow dots at (0,1), (0,-1), represents that f(0) has the value as 0, instead of 1 or -1. Such a function is also known as Signum function.

FAQ (Frequently Asked Questions)

1. Is the Absolute Value of a Modulus Function Always Positive?

The absolute value of a modulus function needs to be non-negative. Since the absolute value of a modulus function generally defines the distance between two points, therefore it can be expressed as:

For f(x), where x represents 0,

therefore,

|f(0)| = 0, which isn't a positive, but a non negative value where x ≥ 0

Similarly,

For a negative integer like -6,

f(x) = f(-6)

|f(-6)| = -(-6)

= |-6|

= 6.

Therefore, |x| = -x, where x is ≤ 0, or a non-positive number.

It can be inferred that the absolute value of any modulus function needs to be non-negative always, not necessarily meaning positive.

2. Is Modulus Function Onto?

For f: R → R, the

f(x) = |x|, where x, x≥0 and -x, if x<0.

Therefore, f(-1) = |-1| = 1.

and f(1) = |1| = 1.

But since, -1 not equal to 1, and f(-1) = f(1),

It can be said that f cannot be considered one-one.

Similarly when -1 ∈ R, R represents Real number set.

f(x) = |x|, the value that stays non-negative. Therefore there isn't any value x present in the domain R where f(x) = |x| = -1.

Therefore, f isn't onto.

Hence, it is proved that the modulus function is neither one-one nor onto.