Question

Sketch the graph of the function

\(f\left( x \right) = \left\{ {\begin{aligned}{\left| x \right|}&{{\bf{if}}\,\,\left| x \right| \le 1}\\1&{{\bf{if}}\,\left| x \right| > 1}\end{aligned}} \right.\)

Short answer

The function \(f\left( x \right)\) represents the following functions:

1) \(f\left( x \right) = \left| x \right|\) for \( - 1 \le x \le 1\)

2) \(f\left( x \right) = 1\) for \(x < - 1\) and \(x > 1\)

Function \(f\left( x \right) = \left\{ {\begin{aligned}{\left| x \right|}&{{\rm{if}}\,\,\,\left| x \right| \le 1}\\1&{{\rm{if}}\,\,\,\left| x \right| > 1}\end{aligned}} \right.\) in thepiecewise formcan be written as shown below:

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

Step-by-step solution

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

The function \(f\left( x \right)\) represents the following functions:

1) \(f\left( x \right) = \left| x \right|\) for \( - 1 \le x \le 1\)

2) \(f\left( x \right) = 1\) for \(x < - 1\) and \(x > 1\)

Function \(f\left( x \right) = \left\{ {\begin{aligned}{\left| x \right|}&{{\rm{if}}\,\,\,\left| x \right| \le 1}\\1&{{\rm{if}}\,\,\,\left| x \right| > 1}\end{aligned}} \right.\) in thepiecewise formcan be written as shown below:

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

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

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