Let $\displaystyle f : \mathbb{R} \rightarrow [-1,1] $ be defined by $\displaystyle f(x)= sin x$
I know there is one as $\displaystyle sin^{-1} x$ exists
I also know that sin x is a surjective function, however i am lost where to go next
Let $\displaystyle f : \mathbb{R} \rightarrow [-1,1] $ be defined by $\displaystyle f(x)= sin x$
I know there is one as $\displaystyle sin^{-1} x$ exists
I also know that sin x is a surjective function, however i am lost where to go next
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)?