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]}