How do you determine if #f(x)= sin^-1 (x)# is an even or odd function?
1 Answer
odd.
Explanation:
Consider any odd function:
# y=g(x) #
Then taking its inverse:
# \ \ \ \ \ \ \ \ \ \ x = g^(-1)(y) #
# :. -x = -g^(-1)(y) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ # (multiplying by#-1# ) ..... [A]
Now as
# \ \ \ \ \ g(-x) = -g(x) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ # (definition of odd fn)
# :. g^(-1)(g(-x)) = g^(-1)(-g(x)) \ # (taking inverses)
# :. -x = g^(-1)(-g(x)) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ # (defn of inverse)
# :. -x = g^(-1)(x) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ # (using [A])
Hence,
Hence the inverse of an odd function is itself an odd function. We kn ow that
We can confirm this graphically:
graph{sinx [-5, 5, -5, 5]}
graph{arcsin(x) [-5, 5, -5, 5]}
