The Modulus Function

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What is a modulus function?

The modulus of any number gives us the magnitude of that number. Using the modulus operation, we can define the modulus function as follows:

\[f\left( x \right) = \left| x \right|\]

Now, if x is non-negative,then the output of f will be x itself. If x is negative,then the output of f will be the magnitude of x, which we can write as \( - x\) (note that if x is negative,then \( - x\) will be positive). Thus, we can redefine the modulus function as follows:

\[f\left( x \right) = \left\{ \begin{array}{l}\;\;\;x,\,\,\, & if\,x \ge 0\\ - x, & if\,x < 0\end{array} \right.\]

The graph of f is plotted below:

Modulus Function - Graph 1

Note that for positive values of x,the graph is the line \(y = x\); for negative values of x, the graph is the line \(y =  - x\). This is in line with the piecewise definition of the modulus function.

Since we can apply the modulus operation to any real number, the domain of the modulus function is \(\mathbb{R}\). The range is clearly the set of all non-negative real numbers, or \(\left( {0,\infty} \right)\).

Example 1: A function f is defined on  \(\mathbb{R}\) as follows:

\[f\left( x \right) = \left\{\!\!\!\begin{gathered}\frac{{\left| x \right|}}{x},\;\;x\ne 0\\\;\;0,\;\;\;\;\;  x = 0\end{gathered} \right.\]

Plot the graph of f.

Solution: When x is positive, we have:

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

When x is negative, we have:

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

Thus, we can redefine f as follows:

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

The graph of f is now easy to plot

Modulus Function - Graph 2

Note the solid dot at \(\left( {0,0} \right)\), and the two hollow dots at \(\left( {0,1} \right)\) and \(\left( {0, - 1} \right)\). This indicates that\(f\left( 0 \right)\) has the value 0 rather than 1 or \( - 1\).

This particular function is known as the  signum  function.