The purpose of this assignment is to develop your ability to formulate and
communicate mathematical arguments. Your complete assignment should have your name and
section number on each page, be stapled, and be neat and legible. Unreadable work will receive no
credit.
You should provide well-written, complete answers to each of the questions. We will look for
correct mathematical arguments, complete explanations, and correct use of English. Your solution
should be formulated in complete sentences. As appropriate, you may want to include diagrams or
equations written out on a separate line. You may read your textbook to find examples of how we
communicate mathematics.
Students are encouraged to use word-processing software to produce high quality solutions.
However, you may find that it is simpler to add graphs and equations using pen or pencil.
1. (2 points) Let L1 (x) = mx + b where m 6= 0. Let L2 (x) be the inverse function of L1 (x).
(a) Find L2 (x).
(b) If m 6= 1, −1, then show that the graphs of L1 (x) and L2 (x) intersect at the point
−b
−b
(
,
).
m−1 m−1
2. (8 points) 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).
In the following problems, use the results of Problem 1, the formula for linearization in Section
4.1, and the formula for the derivative of the inverse of a function in Section 3.8 of the
textbook. Figure 1 of Section 3.8 might help you to 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 0 (a) 6= ±1, explain why the graphs of L1 (x) and L2 (x) intersect on the line given by
y = x.