How do I find the Energy Density Function of $g(t)$ if i am not given an input or impulse response?
If you're referring to energy spectral density, then you need to compute the squared magnitude of the Fourier transform of $g(t)$:
$$\big|G(\omega)\big|^2=\left|\int_{-\infty}^{\infty}g(t)e^{-j\omega t}dt\right|^2\tag{1}$$