Definition of the gamma function for non-integer negative values The gamma function is defined as 
$$\Gamma(x)=\int_0^\infty t^{x-1}e^{-t}dt$$
for $x>0$.
Through integration by parts, it can be shown that for $x>0$,
$$\Gamma(x)=\frac{1}{x}\Gamma(x+1).$$
Now, my textbook says we can use this definition to define $\Gamma(x)$ for non-integer negative values. I don't understand why. The latter definition was derived by assuming $x>0$. So shouldn't the whole definition not be valid for any $x$ value less than zero?
P.S. I have read other mathematical sources and most of them explain things in mathematical terms that are beyond my level. It would be appreciated if things could be kept in relatively simple terms.

The integral $\int_0^\infty t^{x-1}e^{-t}\,dt$ is a "representation" of the Gamma function for $x>0$.  That is, for $x>0$
$$\int_0^\infty t^{x-1}e^{-t}\,dt=\Gamma(x)$$
But the Gamma Function exists for all complex values of $x$ provided $x$ is not $0$ or a negative integer.  
The idea of "representing" a function on a subset of the domain of definition is introduced in elementary calculus courses.  For example, we can represent the function $f(x)=\log(1+x)$ as the series
$$f(x)=\log(1+x)=\sum_{n=1}^\infty\frac{(-1)^{n-1}x^n}{n} \tag 1$$ 
which is valid for $-1<x\le 1$.  However, $f(x)$ exists for all $-1<x$.  
