Im confused by my homework question. Please help me figure it out with as much detail as possible so I can learn. Thanks!

(a) We begin by dening d(x) forx > 0 as follows d = m (X) - f 1(it. 1 :5 Show that d(x) satises d(1) = O, and that it is a one-to-one function on the domain (0, 00). It turns out that d(x) has the range (00, 00), but you don't need to show that here, as it requires the use of "improper integrals", which we will learn about later in this course. You can still use the fact that d(x) has the range (00, W) in Question 2(b) below. (b) Given that d(x) is one-to-one (or equivalently, invertible) it has an inverse function r(x) := d'1(x) What is the domain and range of the function rtx)? Show that r(0) = 1 and use the Inverse Function Theorem and Part | of the Fundamental Theorem of Calculus to prove that r(x) satises r' (x) = r(x) for any xER