Inverse function - Newton's Serpentine

[DNAngel]

I am currently doing inverse functions and have come across a really confusing question.

It relates to Newton's Serpentine. For example,

Let f: [a, infinity) -> R, f(x) = 2x/x^2+1

a) Find the smallest value of a so that f^-1 (x) is a function

b) Find the rule for f^-1, stating its domain and range

My main problem are, how would you go about sketching this function without the use of a graphics calculator?

[luken93]

Do you mean ?
If so, you need to restrict the domain so that it is a One to One Function

a) If you look at the graph, you will see a max at x = 1, restricting it at this will then make it a 1:1 function.
Therefore, a = 1

b)













Dom f(x) = [1,, then the range of f^-1(x) = [1,

Ran f(x) = (0,1), then the range of f^-1(x) = (0,1)

This tells us that the inverse function is steeply increasing, hence:

[pi]

My main problem are, how would you go about sketching this function without the use of a graphics calculator?

Break it up into partial fractions (assuming that your function is ) -although this one may be harder to partialise- and then sketch using addition of ordinates, or through other methods (explained in the spesh course).


EDIT: 2500th post