Question

Sketch the graph of the function

\(f\left( x \right) = \left| {x + {\bf{2}}} \right|\)

Short answer

Function \(f\left( x \right) = \left| {x + 2} \right|\) can be written in the piecewise form as shown below:

\(\begin{aligned}f\left( x \right) &= \left| {x + 2} \right|\\ &= \left\{ {\begin{aligned}{x + 2}&{{\rm{if}}\,\,x + 2 \ge 0}\\{ - \left( {x + 2} \right)}&{{\rm{if}}\,\,\,x + 2 < 0}\end{aligned}} \right.\\ &= \left\{ {\begin{aligned}{x + 2}&{{\rm{if}}\,\,x \ge - 2}\\{ - x - 2}&{{\rm{if}}\,\,\,x < - 2}\end{aligned}} \right.\end{aligned}\)

Step-by-step solution

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

Function \(f\left( x \right) = \left| {x + 2} \right|\) can be written in the piecewise form as shown below:

\(\begin{aligned}f\left( x \right) &= \left| {x + 2} \right|\\ &= \left\{ {\begin{aligned}{x + 2}&{{\rm{if}}\,\,x + 2 \ge 0}\\{ - \left( {x + 2} \right)}&{{\rm{if}}\,\,\,x + 2 < 0}\end{aligned}} \right.\\ &= \left\{ {\begin{aligned}{x + 2}&{{\rm{if}}\,\,x \ge - 2}\\{ - x - 2}&{{\rm{if}}\,\,\,x < - 2}\end{aligned}} \right.\end{aligned}\)

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

The graph of \(f\left( x \right)\) is the line \(y = - x - 2\) for \(x < - 2\) and \(y = x + 2\) for \(x \ge - 2\).

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