Question

In Problems 15–22 determine whether the given set of functions is linearly independent on the interval$\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{\infty }\mathbf{)}$

${\mathbf{f}}_{\mathbf{1}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathbf{+}\mathbf{x}\mathbf{,}\mathbf{}{\mathbf{f}}_{\mathbf{2}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathbf{+}\mathbf{\left|}\mathbf{x}\mathbf{\right|}$

Short answer

The given set of function is linearly dependent

Step-by-step solution

Step 1: Use wronskian and drive the equation

We have set of functions

${\mathrm{f}}_1\left(\mathrm{x}\right)=2+\mathrm{x},{\mathrm{f}}_2\left(\mathrm{x}\right)=2+\left|\mathrm{x}\right|=2+\mathrm{x}\mathrm{and}{\mathrm{f}}_3\left(\mathrm{x}\right)=2+\left|\mathrm{x}\right|=2-\mathrm{x}$

Abd we have to determine if this set is linearly independent on the interval or not as the following method

First, we have to obtain the Wronskian of this set of functions as

$\begin{array}{rcl}\mathrm{W}\left({\mathrm{f}}_1,{\mathrm{f}}_2,{\mathrm{f}}_3\right)& =& \left|\begin{array}{ccc}{\mathrm{f}}_1& {\mathrm{f}}_2& {\mathrm{f}}_3\\ {\mathrm{f}\text{'}}_1& {\mathrm{f}\text{'}}_2& {\mathrm{f}\text{'}}_3\\ {\mathrm{f}}_1\text{'}\text{'}& {\mathrm{f}}_2\text{'}\text{'}& {\mathrm{f}}_3\text{'}\text{'}\end{array}\right|\\ & =& \left|\begin{array}{ccc}2+\mathrm{x}& 2+\mathrm{x}& 2-\mathrm{x}\\ 1& 1& -1\\ 0& 0& 0\end{array}\right|\end{array}$

Second we have to find the determinate of this matrix as

$\begin{array}{rcl}\mathrm{W}\left({\mathrm{f}}_1\left({\mathrm{f}}_1,{\mathrm{f}}_2,{\mathrm{f}}_3\right)\right.& =& (2+\mathrm{x})\times \left|\begin{array}{cc}1& -1\\ 0& 0\end{array}\right|-(2+\mathrm{x})\times \left|\begin{array}{cc}1& -1\\ 0& 0\end{array}\right|+(2-\mathrm{x})\times \left|\begin{array}{cc}1& 1\\ 0& 0\end{array}\right|\\ & =& (2+\mathrm{x})\left[\right(1\left)\right(0)-(0\left)\right(-1\left)\right]-(2+\mathrm{x})\left[\right(1\left)\right(0)-(0\left)\right(-1\left)\right]+(\mathrm{x}-2)\left[\right(1\left)\right(0)-(0\left)\right(1\left)\right]\\ & =& 0\\ & & \end{array}$

Since the determinate of the wronskian of the given set of functions equal Zero,

Then this set of function is linearly dependent.