# Math Help - How many right inverses does the function have?

1. ## How many right inverses does the function have?

Let $f : \mathbb{R} \rightarrow [-1,1]$ be defined by $f(x)= sin x$

I know there is one as $sin^{-1} x$ exists
I also know that sin x is a surjective function, however i am lost where to go next

2. Rephrasing the question:

What are the functions g : [-1, 1] to R such that
f(g(x)) = x (or sin(g(x)) = x)?

g = sin^-1(x) is one solution
What about g(x) = sin^-1(x) + 2*pi?
What about g(x) = sin^-1(x) + 4*pi?
What about g(x) = -sin^-1(x) + pi?
What about g(x) = -sin^-1(x) + 3*pi?

Also, the function does not even need to be continuous. What about the function
g(x) = sin^-1(x) + 2*pi for 0 <= x <= 1
and g(x) = sin^-1(x) for -1 <= x < 0

Can you think of the most general form for g(x)?