Answer True or False to each statement below and provide a justification for your answer.

$($a$)x=-\frac{1}{3}$ is in the domain of $f(x)=\frac{x-2}{3x+1}.$

$($b$)x=-3$ is in the domain of $f(x)=\sqrt[\phantom{2}]{x+3}.$

$($c$)x=-4$ is in the domain of $f(x)=\sqrt[\phantom{2}]{x+3}.$