Determine the domain of each of the following functions. Give your answer in interval notation as well as sketching a graph of the real number line showing the domain graphically.

$($a$)g(x)=x^2$

$($b$)f(x)=\sqrt[\phantom{2}]{3x-7}$

$($c$)h(x)=\frac{5}{x^2-25}$

$($d$)f(x)=\sqrt[\phantom{2}]{x^2-4x+3}$

$($e$)g(x)=\frac{\sqrt[\phantom{2}]{x^2-4x+3}}{x^3+8}$