Correctly integrating the product of two functions I'm trying to determine the probability of finding a function inside another function.
While plotting the results do give me a sign of there being an actual non-zero number, integrating to determine the number gives me zero.
Plotting these two functions gives me
$$ f(x) = \left( \frac{1}{\pi } \right)^{1/4} e^{- \frac{1}{2 } x^2} $$
$$ f_{1}(x) = \left( \frac{1}{ 2\pi} \right)^{1/4}   x\ e^{- \frac{1}{4} x^2} $$

Which obviously shows that a portion of is found inside the other function.
When integrating over all space defined as 
$$ \int_{-\infty}^{\infty} f_{1}^{*}(x)\ f(x) dx $$
(in which after integrating, I take the square modulus to determine the probability)
$$ \int_{-\infty}^{\infty} \left( \frac{1}{2\pi} \right)^{1/4} \left( \frac{1}{\pi }\ \right)^{1/4} e^{ -\frac{3}{4 } x^2}  x\ dx = 0 $$
Is this due to me integrating over all space? In which judging by my plot, I should change my bounds to 0 to $\infty$ instead?

Your orange plot is incomplete.  If you were to plot your $f_1$ over the interval $[-15,15]$ you would see that it has a negative lobe that is symmetric with the positive lobe you plotted.  There are two ways to fix this:
1)  Do what you suggest and only integrate over the intersection of the supports of the two functions.  In this case, $[0,\infty)$.
2) Actually write down your $f_1$'s complete definition:  $$f_1(x) = \begin{cases} \frac{ \left( \frac{m \omega}{\pi \hbar} \right)^{1/4} }{\sqrt{\hbar m \omega}}\  x\ e^{ \frac{m \omega}{4 \hbar} x^2} ,& x \geq 0 \\ 0 ,& x < 0  \end{cases} \text{,}  $$ so that the product of your functions is actually zero to the left of zero.
