#### To determine

**To find:** Graph of given function

#### Answer

The given function is one to one.

Derivative is 11−y2

#### Explanation

You can determine whether or not the given function is one to one by just looking at the graph with the help of Horizontal Line Test. Horizontal Line Test says that a function f is one to one if and only if any horizontal line intersects the graph of given function exactly at one point. In other words, the Horizontal Line Test can be stated as:

A function f is NOT one to one if and only if there exists a horizontal line which intersects the graph of given function at least at two points.

The following figure shows that when we restrict our domain to -π2,π2, then sinx is one to one.

Any horizontal line will intersect the given graph at one point only. Therefore, by using the Horizontal Line Test, we can conclude that the given function is one to one.

Now, we need to compute the derivative of f-1 by Note 2.

Given f-1x=sin-1x.Write y=f-1x=sin-1x⇒siny=x.

Now, differentiating both sides with respect to x, we get

cosydydx=1⇒dydx=1/cosy.

Since cos2y+sin2y=1, then cosy=±1−sin2y but here cosy>0 on the given interval, we have

dydx=11−sin2y=11−y2

**Conclusion:** The given function is one to one.

Derivative is 11−y2