Expert: David Montiel - 10/21/2009

Question
I REALLY need to know how to go about finding the inverse of a transcendental function. I have no idea how to approach it and it may be on a test tomorrow. please any tips.

f(x)= 1 + x^2 + tan ( (pi*x) / 2)     how do i find the inverse??

Help will be much appreciated thanks.





P.s. in taking the infinate limit of a trig function (eg sinx) would it be correct to say it varies and does not exist) like the limit x>INFINITY of sinx/x would be 0?

Answer
Hello Rick,

Every transcendental function has it's inverse. In the case of trigonometric functions you have asin, acos, atan (inverse functions of sin, cos, tan, etc). For instance if you have

y=f(x)=tan(x) then the inverse function is x=g(y)=asin(y), right?

BTW asin is also called arcsin or sin^-1, same for acos, etc. (you can go here to learn more:
http://en.wikipedia.org/wiki/Inverse_trigonometric_functions )

For the equation you write, I don't think you can find the inverse in analytical form because you have a combination of a transcendental function (tan) and a polynomial (x^2+1)... so don't worry I don't think that one will be on the test :)

Now, about the limit of sin(x) as x>INFINITY you are right, it does not exist, because the function does not converge asymptotically to any value... so you definitely get the picture.

Hope this helps and good luck tomorrow.

Cheers,
David