Question

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

Answer 1

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]}