1. Conslaer the Curve y =JX) = 2"-1.

Question

A. Find the exact area of the region in the first quadrant bounded by the curves y = fx) = 2-1
and y = x. ("Exact area" means no calculator numbers.)
B. Find the inverse function y =flx).
C. Using part A and the notion of symmetry between a function and its inverse, find the exact
area of the region in the first quadrant bounded by the curves y = flx) and y = x. Explain your
reasoning. (Hint: Think "graphically" and little or no math will need to be done!)

Answer 1

No idea what y =JX) = 2"-1 is.
If you mean y = 2^x - 1
then
(A) the area on the interval [1,2] is
∫[1,2] x-(2^x - 1) dx
(B) f^-1(x) = log₂(x+1)
(C) see what you can do with this.
If you actually meant y = 2^(x-1) then make the appropriate adjustments