5.3 Modulus Function

The function $f(x)$ defined by, $$f(x) = \left| x \right| = \left\{ \matrix{x\;\;\;\;\;,\;when\;x \ge 0 \hspace{1em} \cr - x\,\;\;,\;when\;x < 0 \hspace{1em} \cr} \right.$$ is called the modulus function.

To plot the graph, just divide it into two parts.

First one is when $x>0$. We consider the first function i.e., $y=x$ means a straight line passing through origin and positive slope $+1$. But the graph is valid only when $x>0$ i.e., only for $I$ and $IV$ quadrant. So just remove the straight line part which lies in $III$ quadrant.

Similarly, for second part when $x<0$. We consider the second function i.e., $y=-x$ means a straight line passing through origin and negative slope $+1$. But the graph is valid only when $x<0$ i.e., only for $II$ and $III$ quadrant. So just remove the straight line part which lies in $IV$ quadrant.

The remaining part of straight line represents the graph of mod function as shown in the figure.

Using the above concept, if we have to plot for the function $y = - \left| x \right|$, we can directly change the sign as shown \[f(x) = - \left| x \right| = \left\{ \begin{gathered}- x\;\;\;\;\;,\;when\;x \geqslant 0 \hspace{1em} \\ x{\mkern 1mu} \;\;{\text{ }},\;when\;x < 0 \hspace{1em} \\ \end{gathered} \right.\]

Example 14. Plot the graph of a function $f(x) = \left| {x - {k_1}} \right| + {k_2}$.

Solution: To plot the graph of the function $$y = \left| {x - {k_1}} \right| + {k_2}$$ First of all bring the constant term to the left side, $$y - {k_2} = \left| {x - {k_1}} \right|$$ Now, assume both left and right side equals to $0$ separately i.e.,