How do you determine if f(x)= sin^-1 (x) is an even or odd function?

1 Answer
Oct 19, 2017

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 is odd so we have:

\ \ \ \ \ 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, g^(-1)(x) is an odd function.

Hence the inverse of an odd function is itself an odd function. We kn ow that sinx is an odd function and hence its inverse sin^(-1)x is also an odd function.

We can confirm this graphically:

graph{sinx [-5, 5, -5, 5]}

graph{arcsin(x) [-5, 5, -5, 5]}