# Inverse Function of sine

1. Dec 3, 2011

### ZedCar

1. The problem statement, all variables and given/known data
Determine if the following function is invertible.
If it is, find the inverse function.

f (x) = -sin(-x)
-∏ < x < ∏

3. The attempt at a solution

f(x) = -sin(-x)
y = -sin(-x)
-(sin y)^-1 = -x
(sin y)^-1 = x

when x = -∏, y = 0
when x = ∏, y = 0

f^-1(x) = (sin x)^-1
0 = x = 0

2. Dec 3, 2011

### ZedCar

Actually, now I'm wondering, is the answer simply that the function is not monotonic and therefore it is not invertible?

3. Dec 3, 2011

### Staff: Mentor

The interval doesn't include $\pi$ or $-\pi$
Have you sketched a graph of this function? If you know the graph of y = f(x), you can get the graph of y = -f(-x) by doing a couple of reflections. Having a graph should give you a good idea of whether your function has an inverse.

What rule or theorem are you using to determine whether a function has an inverse?

4. Dec 3, 2011

### ZedCar

Yes, I wasn't sure what to do about that. On other questions I've tried it usually states, for example, -∏ <= x <= ∏, but in this example it didn't have an equals in the interval.

I've just input it into http://rechneronline.de/function-graphs/ and it appears that it is not monotonic, and therefore not invertible.

I wasn't really using any specific rule for determining if the function is monotonic, as so far the functions I've been trying aren't too difficult eg (X^2 - 5) or (3x + 3).

Below is the method I've been using to determine the inverse functions.

For example,

f(x) = x^2 - 5 0 <= x < ∞
y = x^2 - 5
(y + 5)^0.5 = x
when x = 0, y = -5

therefore f^-1(x) = (x + 5)^0.5
when x = 0, y = -5

Is there a specific rule for determining if a function is monotonic?