Function Properties: One-To-One

We can numerically check if a function is one-to-one or not as follows:

1. Set up an equation of the form $f(x_1) = f(x_2)$. For instance, for $f(x) = 2x + 3$, we will do: $2x_1 + 3 = 2x_2 + 3$.
2. Solve the equation for $x_1$. We do the following:
   $\begin{align} 2x_1 + 3 &= 2x_2 + 3\ [\text{Original equation}]\\ 2x_1 &= 2x_2\ [\text{Subtracting 3 from each side}]\\ x_1 &= x_2\ [\text{Dividing each side by 2}]. \end{align}$
3. If you got $x_1 = x_2$ as the only solution to the equation, then the function $f$ is one-to-one. In our example for $f(x) = 2x + 3$, we indeed found that $f$ is injective.

Contrastingly, the function $f(x) = x^2$ is not injective because we get $(x_1)^2 = (x_2)^2 \Rightarrow x_1^2 = x_2^2 \Rightarrow x_1 = \pm x_2$. In particular, because we got a solution that isn't $x_1 = x_2$ (we also got $x_1 = -x_2$,) this means that $f(x) = x^2$ is not injective.