# Does this function ever have the same value twice

complex-analysis

Consider the function from a real value to a complex value:

$$f(x) = \cos(\sqrt{2} x ) + i \sin(\sqrt{3} x)$$

My contention is it never has the same complex value for two different real values of $$x$$.

i.e. $$f(x)=f(y) \implies x=y$$

Is this true? Is there a proof?

My second contention is that it has no definable inverse. $$f^{-1}(z)$$ from a complex number to a real number.

Edit:

I realised this must be wrong! But I think it is correct for this function to a quaternion:

$$g(x) = \cos(\sqrt{2} x ) + i \sin(\sqrt{2} x) + j\cos(\sqrt{3} x ) + k \sin(\sqrt{3} x)$$

The result is true for your $$g(x)$$.
For real numbers $$u, v$$, if $$\cos(u) = \cos(v)$$ and $$\sin(u) = \sin (v)$$, then we must have $$u - v \in 2\pi \Bbb Z$$.
Thus if $$x, y$$ are different real numbers such that $$g(x) = g(y)$$, then we must have $$ax - ay \in 2\pi \Bbb Z$$ and $$bx - by \in 2\pi \Bbb Z$$, where $$a = \sqrt 2$$ and $$b = \sqrt 3$$.
Taking quotient of the two, we get $$a/b \in \Bbb Q$$ which is false.
It is however not true for your $$f(x)$$. Take e.g. $$x = -y = \frac \pi {\sqrt 3}$$.