Suppose that f(x) is a differentiable function and assume that g(x) is the inverse function of f(x). Let L1(x) be the linearization of f(x) at x = a and let L2(x) be the linearization of g(x) at x = b where b = f(a). visualize what is going on geometrically.
(a) Write the formulas for L1(x) and L2(x).
(b) How are the slopes of L1(x) and L2(x) related?
(c) If the graph of L1(x) is not a horizontal line, then show that L2(x) is the inverse function of L1(x). (It is helpful to use that b = f(a) and a = g(b) in this problem.)
(d) If f'(a) doesn't equal A?±1, explain why the graphs of L1(x) and L2(x) intersect on the line given by y = x