Question

Sketch the graph of the function

\(g\left( x \right) = \left| {\left| x \right| - {\bf{1}}} \right|\)

Short answer

Function \(g\left( x \right) = \left| {\left| x \right| - 1} \right|\) can be written in thepiecewise formas shown below:

\(\begin{aligned}g\left( x \right) &= \left\{ {\begin{aligned}{\left| x \right| - 1}&{{\rm{if}}\;\left| x \right| - 1 \ge 0}\\{ - \left( {\left| x \right| - 1} \right)}&{{\rm{if}}\,\,\,\,\left| x \right| - 1 < 0}\end{aligned}} \right.\\ &= \left\{ {\begin{aligned}{\left| x \right| - 1}&{{\rm{if}}\,\,\,\left| x \right| \ge 1}\\{ - \left| x \right| + 1}&{{\rm{if}}\,\,\,\left| x \right| < 1}\end{aligned}} \right.\\ &= \left\{ {\begin{aligned}{x - 1}&{\left| x \right| \ge 1\,\,\,\,{\rm{and}}\,x \ge 0}\\{ - x - 1}&{\left| x \right| \ge 1\,\,\,\,{\rm{and}}\,x < 0}\\{ - x + 1}&{\left| x \right| < 1\,\,\,\,{\rm{and}}\,x \ge 0}\\{ - \left( { - x} \right) + 1}&{\left| x \right| < 1\,\,\,\,{\rm{and}}\,x < 0}\end{aligned}} \right.\\ &= \left\{ {\begin{aligned}{x - 1}&{x \ge 1}\\{ - x - 1}&{x \le - 1}\\{ - x + 1}&{0 \le x < 1}\\{x + 1}&{ - 1 < x < 0}\end{aligned}} \right.\end{aligned}\)

Step-by-step solution

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

Function \(g\left( x \right) = \left| {\left| x \right| - 1} \right|\) can be written in thepiecewise formas shown below:

\(\begin{aligned}g\left( x \right) &= \left\{ {\begin{aligned}{\left| x \right| - 1}&{{\rm{if}}\;\left| x \right| - 1 \ge 0}\\{ - \left( {\left| x \right| - 1} \right)}&{{\rm{if}}\,\,\,\,\left| x \right| - 1 < 0}\end{aligned}} \right.\\ &= \left\{ {\begin{aligned}{\left| x \right| - 1}&{{\rm{if}}\,\,\,\left| x \right| \ge 1}\\{ - \left| x \right| + 1}&{{\rm{if}}\,\,\,\left| x \right| < 1}\end{aligned}} \right.\\ &= \left\{ {\begin{aligned}{x - 1}&{\left| x \right| \ge 1\,\,\,\,{\rm{and}}\,x \ge 0}\\{ - x - 1}&{\left| x \right| \ge 1\,\,\,\,{\rm{and}}\,x < 0}\\{ - x + 1}&{\left| x \right| < 1\,\,\,\,{\rm{and}}\,x \ge 0}\\{ - \left( { - x} \right) + 1}&{\left| x \right| < 1\,\,\,\,{\rm{and}}\,x < 0}\end{aligned}} \right.\\ &= \left\{ {\begin{aligned}{x - 1}&{x \ge 1}\\{ - x - 1}&{x \le - 1}\\{ - x + 1}&{0 \le x < 1}\\{x + 1}&{ - 1 < x < 0}\end{aligned}} \right.\end{aligned}\)

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

The graph of the function \(f\left( x \right) = \left\{ {\begin{aligned}{x - 1}&{x \ge 1}\\{ - x - 1}&{x \le - 1}\\{ - x + 1}&{0 \le x < 1}\\{x + 1}&{ - 1 < x < 0}\end{aligned}} \right.\) is shown below.