Answer
\[s\,\left( x \right) = \left\{ \begin{gathered}
1\,\,\,\,\,\,\,\,if\,\,\,\,\,\,\,\,\,x > 0 \hfill \\
\hfill \\
- 1\,\,\,\,\,\,\,if\,\,\,\,\,\,\,x < 0 \hfill \\
\end{gathered} \right.\]
Work Step by Step
\[\begin{gathered}
Function\,\,f\,\left( x \right) = \left| x \right|\,\,can\,be\,\,\,\,written\,\, \hfill \\
{\text{using the definiton of absolute value}} \hfill \\
\hfill \\
f\,\left( x \right) = \left\{ \begin{gathered}
x\,\,\,\,\,\,\,\,if\,\,\,\,\,\,\,\,x <0 \hfill \\
\hfill \\
- x\,\,\,\,\,if\,\,\,\,\,\,\,\,x < 0 \hfill \\
\end{gathered} \right. \hfill \\
\hfill \\
since\,\,the\,slope\,of\,\,the\,line\,f\,\left( x \right) = x\,is\,1\,\,and\,slope \hfill \\
of\,the\,line\,f\,\left( x \right) = - x\,\,is\,\, - 1\,,\,slope\,function\,is\,given\,\,by \hfill \\
\hfill \\
s\,\left( x \right) = \left\{ \begin{gathered}
1\,\,\,\,\,\,\,\,if\,\,\,\,\,\,\,\,\,x > 0 \hfill \\
\hfill \\
- 1\,\,\,\,\,\,\,if\,\,\,\,\,\,\,x < 0 \hfill \\
\end{gathered} \right. \hfill \\
\hfill \\
\end{gathered} \]
