Question

is sin(sin^-1x)=sin-1(sinx) an identical? why or why not?

Answer 1

The issue is domains. The domain of the function on the right is the closed interval
[-1,1]

But the domain of the function on the left is all the real numbers.

For numbers which lie in both of these domains, both functions are the identity function. That is
\[ \sin(\arcsin x) = \arcsin(\sin x) = x \]

Answer 2

so not identical

Answer 3

No. Technically, the definition of a function includes its domain. Hence two functions, f and g, are identical if
- both functions have exactly equal domain, call it \( D \)
- for every \( x \in D \), \( f(x) = g(x) \)

For this pair of functions the second condition is met but not the first.

Answer 4

Questions like this are a favorite on first-year calculus exams.