Question

Sketch the graph of the function

\(g\left( t \right) = \left| {{\bf{1}} - {\bf{3}}t} \right|\)

Short answer

Function \(g\left( t \right) = \left| {1 - 3t} \right|\) can be written in the piecewise form as shown below:

\(\begin{aligned}g\left( x \right) &= \left| {1 - 3t} \right|\\ &= \left\{ {\begin{aligned}{1 - 3t}&{{\rm{if}}\,\,1 - 3t \ge 0}\\{ - \left( {1 - 3t} \right)}&{{\rm{if}}\,\,\,1 - 3t < 0}\end{aligned}} \right.\\ &= \left\{ {\begin{aligned}{1 - 3t}&{{\rm{if}}\,\,t \le \frac{1}{3}}\\{3t - 1}&{{\rm{if}}\,\,\,t > \frac{1}{3}}\end{aligned}} \right.\end{aligned}\)

Step-by-step solution

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

Function \(g\left( t \right) = \left| {1 - 3t} \right|\) can be written in the piecewise form as shown below:

\(\begin{aligned}g\left( x \right) &= \left| {1 - 3t} \right|\\ &= \left\{ {\begin{aligned}{1 - 3t}&{{\rm{if}}\,\,1 - 3t \ge 0}\\{ - \left( {1 - 3t} \right)}&{{\rm{if}}\,\,\,1 - 3t < 0}\end{aligned}} \right.\\ &= \left\{ {\begin{aligned}{1 - 3t}&{{\rm{if}}\,\,t \le \frac{1}{3}}\\{3t - 1}&{{\rm{if}}\,\,\,t > \frac{1}{3}}\end{aligned}} \right.\end{aligned}\)

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

The graph of \(g\left( t \right)\) is the line \(g\left( t \right) = 1 - 3t\) for \(t < \frac{1}{3}\) and \(g\left( t \right) = 3t - 1\) for \(t > \frac{1}{3}\).

The graph of the function \(g\left( t \right)\) is shown below.