3.7 Modulus function

The modulus function returns positive value of a variable or an expression. For this reason, this function is also referred as absolute value function. In reference to modulus of an independent variable, the function results in a non-negative value of the variable, irrespective of whether independent variable is positive or negative. The intent of this expression for different values of “x” is expressed as :

| x, x≥0 |x| = || -x, x<0

When value of "x" is a non-negative number, then function is "x"; otherwise "-x". The negative sign in the second interval ensures that the function return a positive value when variable is negative.

There are different interpretations of modulus, depending on situations. In particular, modulus of a real variable and modulus of an expression needs to be interpreted in proper context. Modulus of real variable is equivalent to modulus of real number as variable can take real values only. On the other hand modulus of an expression needs to be treated differently. An expression in a variable is a function. The fact that modulus modifies function values changes function properties. Besides, modulus function can also be interpreted to represent distance of a point with respect to a reference point.

In order to investigate the nature of modulus function, we investigate its plot with respect to independent variable. Here, we calculate few initial values to draw the plot as :

$$Forx=-\mathrm{2,}y=\left|x\right|=-x=-\left(-2\right)=2$$

$$Forx=-\mathrm{1,}y=\left|x\right|=-x=-\left(-1\right)=1$$

$$Forx=\mathrm{0,}y=\left|x\right|=x=0$$

$$Forx=\mathrm{1,}y=\left|x\right|=x=1$$

$$Forx=\mathrm{2,}y=\left|x\right|=x=2$$

The graph of the function is shown here :

The graph of modulus function is continuous having a corner at x=0. It means that graph is differentiable at all points except at x=0 (we can not draw a tangent at a corner). Since graph is symmetric about y-axis, modulus function is even function. We also see that there are pair of x values for non-zero values of y. This, in turn, means that image and pre-images are not uniquely related. As such, modulus function is not invertible.

It is clear from the graph that the domain of modulus function is "R". However, the function values are only positive values, including zero. Hence, range of modulus function is upper half of the real number set, including zero.

$$Domain=R$$

$$Range=\left[\mathrm{0,}\infty \right)$$

In the nutshell, we see that modulus represents a non-negative value. Going by this interpretation, modulus of a variable can also be thought as the square root of the square of the variable. We must emphasize that an unsigned square root is a non-negative value same as modulus value. Hence,

$$\left|x\right|=\sqrt{x^2}$$

Modulus function is a non – negative value. There are some other such non-negative expressions. Here, we enumerate them for ready reference :

• Modulus of an expression or variable : |x|
• Even powers of an expression or variable : $x^{2n},\text{where}n\in Z$
• Even root of an expression or variable : $x^{\frac{1}{2n}},\text{where}n\in Z$
• y = 1 – sinx; y = 1 – cosx; (as sinx ≤1 and cosx ≤1)

Modulus and equality

Modulus of a variable or an expression is referred frequently in different mathematical contexts. Modulus is also widely used to denote intervals. The representation of interval, in terms of modulus, has the advantage of compactness. We, however, need to be careful in interpreting modulus. Here, we present certain important interpretations of expressions and equations, which involve modulus.