Answer
See answer below
Work Step by Step
We are given:
$f_1(x)=x^{2}$ for $x \geq 0$
$f_1(x)=3x^3$ for $x \lt 0$
$f_2(x)=7x^{2}$
and $I=(-\infty, +\infty)$
We take $f'_1(x)=9x^2$
and $f'_2x(x)=14x$
The augmented matrix of this system is:
$W[f_1,f_2,f_3](-1)=\begin{bmatrix}
-3 & 7\\
9& -14
\end{bmatrix}=-3\times(-14)-7\times9=-21$ for all $x$ in $(-\infty, \infty)$
Since $W=-21 \ne 0 $, the set of vectors $\{f_1, f_2, f_3 \}$ is linearly independent on $(-\infty, +\infty)$
