Find the inverse of f(x)=(1+tan(x))^(3/2)?

Question

this is for violet h.......... i would never get this on the exam if i never learn how to do it..

Answer 1

do it yourself! the whole of yahoo answers won't be there when you're sitting in that exam room.

Answer 2

It shouldn't have an inverse, because graphically it fails the horizontal line test.

However, if we were to solve for the hypothetical inverse, we would first make f(x) into y.

y = (1 + tan(x))^(3/2)

Swap the x and y terms.

x = (1 + tan(y))^(3/2)

And now, solve for y.

Take both sides to the power of (2/3), to get

x^(2/3) = 1 + tan(y)

Solve for y.

x^(2/3) - 1 = tan(y)

Take the inverse tan of both sides, or arctan.  We are not allowed to do this unless we assume tan(y) is between -pi/2 and pi/2, so that's what we will assume.

tan(y) = x^(2/3) - 1

So

y = arctan (  x^(2/3) - 1 )

So under assumptions,

f^(-1)(x) = arctan (  x^(2/3) - 1 )