Existence, Uniqueness and Counterexamples · Hard problem

There is exactly one real $x$ with $x^3 = x$.

There is exactly one positive real $x$ with $x^x = \tfrac12$.

There is exactly one real $x$ with $e^x = 1 + x$.

There is exactly one real $x$ with $\sin x = \tfrac{x}{10}$.