How to determine if a function is one-to-one?
To show that $f$ is 1-1, you could show that $$f(x)=f(y)\Longrightarrow x=y.$$ So, for example, for $f(x)={x-3\over x+2}$:
Suppose ${x-3\over x+2}= {y-3\over y+2}$. Then: \begin{align*} &{x-3\over x+2}= {y-3\over y+2} \\ \Longrightarrow& (y+2)(x-3)= (y-3)(x+2)\\ \iff& yx+2x-3y-6= yx-3x+2y-6\\ \iff&2x-3y =-3x+2y\\ \iff&2x+3x =2y+3y\\ \iff&5x =5y\\ \iff&x=y \end{align*} So $f(x)={x-3\over x+2}$ is 1-1.
I'll leave showing that $f(x)={{x-3}\over 3}$ is 1-1 for you.
Alternatively, to show that $f$ is 1-1, you could show that $$x\ne y\Longrightarrow f(x)\ne f(y).$$
Or, to show that a differentiable $f$ is 1-1, you could show that its derivative $f'$ is either always positive or always negative.
You would discover that a function $g$ is not 1-1, if, when using the first method above, you find that the equation is satisfied for some $x\ne y$. For example, take $g(x)=1-x^2$. Then
$$ \eqalign{ &g(x)=g(y)\cr \iff&{1-x^2}= {1-y^2} \cr \iff&-x^2= -y^2\cr \iff&x^2=y^2\cr} $$ The above equation has $x=1$, $y=-1$ as a solution. So, there is $x\ne y$ with $g(x)=g(y)$; thus $g(x)=1-x^2$ is not 1-1.
Of course, to show $g$ is not 1-1, you need only find two distinct values of the input value $x$ that give $g$ the same output value.
Although you rightfully point out that the graphical method is unreliable; it is still instructive to consider the methods used and why they work:
Graphically, you can use either of the following:
Use the "Horizontal Line Test":
$f$ is 1-1 if and only if every horizontal line intersects the graph of $f$ in at most one point. Note that this is just the graphical interpretation of "if $x\ne y$ then $f(x)\ne f(y)$"; since the intersection points of a horizontal line with the graph of $f$ give $x$ values for which $f(x)$ has the same value (namely the $y$-intercept of the line).
Use the fact that a continuous $f$ is 1-1 if and only if $f$ is either strictly increasing or strictly decreasing. This, of course, is equivalent to the derivative being always positive or always negative in the case where $f$ is differentiable. (Note this method applies to only the green function below.)
A one-to-one function is an injective function. A function $f:A\rightarrow B$ is an injection if $x=y$ whenever $f(x)=f(y)$.
Both functions $f(x)=\dfrac{x-3}{x+2}$ and $f(x)=\dfrac{x-3}{3}$ are injective.
Let's prove it for the first one
$$ \begin{eqnarray*} f(x) &=&f(y)\Leftrightarrow \frac{x-3}{x+2}=\frac{y-3}{y+2} \\ &\Rightarrow &\left( y+2\right) \left( x-3\right) =\left( y-3\right) \left( x+2\right) \qquad(\text{for }x\neq-2,y\neq -2)\\ &\Rightarrow &xy-3y+2x-6=xy+2y-3x-6 \\ &\Rightarrow &-3y+2x=2y-3x\Leftrightarrow 2x+3x=2y+3y \\ &\Rightarrow &5x=5y\Rightarrow x=y. \end{eqnarray*}$$
So we concluded that $f(x) =f(y)\Rightarrow x=y$, as stated in the definition.
As for the second, we have $$ \begin{eqnarray*} f(x) =f(y)\Leftrightarrow \frac{x-3}{3}=\frac{y-3}{3} \Rightarrow &x-3=y-3\Rightarrow x=y. \end{eqnarray*} $$
An example of a non injective function is $f(x)=x^{2}$ because $$ \begin{eqnarray*} f(x) =f(y)\Leftrightarrow x^{2}=y^{2} \Rightarrow x=y\quad \text{or}\quad x=-y. \end{eqnarray*} $$
For your modified second function $f(x) = \frac{x-3}{x^3}$, you could note that $$f(x) - f(y) = \frac{(x-y)((3-y)x^2 +(3y-y^2) x + 3 y^2)}{x^3 y^3}$$ As a quadratic polynomial in $x$, the factor $ (3-y)x^2 +(3y-y^2) x + 3 y^2$ has discriminant $y^2 (9+y)(y-3)$. So when either $y > 3$ or $y < -9$ this produces two distinct real $x$ such that $f(x) = f(y)$.