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Problem 57: (Function Notation With Sine)

Consider the function $f:\mathbb{R}\to\mathbb{R}$ defined by $f(x)=\sin x$.

  1. What are the sets $f(\mathbb{R})$ and $f([0,\pi/2))$?
  2. What are the sets $f^{-1}(0)$ and $f^{-1}([0,\pi/2))$?
  3. Find a set $A\subseteq \mathbb{R}$ so that $g:A\to \mathbb{R}$ defined by $g(x)=\sin(x)$ is injective.
  4. Find a set $B\subseteq \mathbb{R}$ so that $h:A\to B$ defined by $h(x)=\sin(x)$ is surjective.

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