Question

Sketch the graph of the function

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

Short answer

Function \(f\left( x \right) = \frac{{\left| x \right|}}{x}\) can be written in the piecewise form as shown below:

\(\begin{aligned}f\left( x \right) &= \frac{{\left| x \right|}}{x}\\ &= \left\{ {\begin{aligned}{ - \frac{x}{x}}&{{\rm{if}}\,\,x < 0}\\{\frac{x}{x}}&{{\rm{if}}\,\,\,x > 0}\end{aligned}} \right.\\ &= \left\{ {\begin{aligned}{ - 1}&{{\rm{if}}\,\,x < 0}\\1&{{\rm{if}}\,\,\,x > 0}\end{aligned}} \right.\end{aligned}\)

Step-by-step solution

Write the function \(f\left( x \right)\) in the piecewise form

Function \(f\left( x \right) = \frac{{\left| x \right|}}{x}\) can be written in the piecewise form as shown below:

\(\begin{aligned}f\left( x \right) &= \frac{{\left| x \right|}}{x}\\ &= \left\{ {\begin{aligned}{ - \frac{x}{x}}&{{\rm{if}}\,\,x < 0}\\{\frac{x}{x}}&{{\rm{if}}\,\,\,x > 0}\end{aligned}} \right.\\ &= \left\{ {\begin{aligned}{ - 1}&{{\rm{if}}\,\,x < 0}\\1&{{\rm{if}}\,\,\,x > 0}\end{aligned}} \right.\end{aligned}\)

Sketch the graph of \(f\left( x \right)\)

The graph of \(f\left( x \right)\) is the line \(y = - 1\) for \(x < 0\) and \(y = 1\) for \(x > 0\).

The graph of the function \(f\left( x \right)\) is shown below.