Can't understand Injective function from the test

I had a test in classroom and I couldn't understand the solution of it.

The question is like this.

Assume that $x\neq y\neq z\neq x$.

Is the function $f :\left \{x, y, z \right \} \rightarrow \left \{ 0, 1 \right \}$ defined by $f(x) = 0,$ $f(y) = 1,$ $f(z) = 0$ injective?

justify your answer.

So what I wrote is that

Since $f(x) = 0$ and $f(z) =0$ therefore not injective.

However I did not get the full mark so I checked the solution and it says

because $f(x) = 0 = f(z)$ and $x\neq z$ there for not injective.

I don't understand why the $x\neq z$ is so important in injective function and would it be injective if $x = z?$.

Thank you in advance.