how to sketch cos^-1(x^2)?

[TrueTears]

Well there is a question that says find the domain and range of cos^-1(x^2).

I found the domain pretty easily which is just  -1<=x^2<=1

solve for x^2<=1 and we get -1<=x<=1. But how do i work out the range? I never learnt how to sketch it, so i can't do it graphically, and i do not know any analytical way of doing it.

[humph]

You certainly need to know how to sketch and other such basic inverse trigonometric functions. In any case, you do get the domain and range of on the formula sheet for the non-calculator exam. Anyway, the range of is . So you use that and the fact that the only difference between the range of and the range of is that the latter won't have any of the images of the interval , as it's impossible for to be a negative number.. You should be able to work out by plugging in these numbers that the image of under is , and so the range of must therefore be .

[TrueTears]

ah i see, so all u pretty much have to do is just plug the domain end points back into the equation and u get the min/max values of the graph?

[humph]

In this case, yes, because it's a one-to-one function, so it must be either non-decreasing or non-increasing. In fact, if you know a function is one-to-one and you're trying to find its minimum and maximum on a closed interval, then these occur at the endpoints of the interval.

[TrueTears]

ah thank you very much humph. I fully understand it now

[humph]

Bear in mind though that this technique doesn't work at all on a function that isn't one-to-one; for example, we have for the function on the interval that , whereas and . In this case, you really need to actually know how to graph or use other techniques like calculus, which can help you find local maxima and minima.