1) This exercise illustrates limitations to the usefulness of graphs. Consider the function

f: R → R,

f (x) = {■(e^((-1)⁄x^2 )&x≠[email protected]&x=0)┤

Why is it reasonable to deﬁne f (0) = 0?

Using technology, construct a table of values, for the inputs {0, 0.1, 0.2, 0.3, 0.4}.

Using your answer to (b), attempt to construct a rough graph of f (x) over the interval

[0, 0.4].

What property of the function makes it so difﬁcult to construct a meaningful graph?

Use technology (graphing calculator or computer algebra system) to graph f. Discuss your results.

2)

Let f be a function. Find a formula for the function whose graph is obtained from the graph of f by shifting two units to the left and then by stretching vertically by a factor of three. Then, ﬁnd the function whose graph is obtained from the graph of f by stretching vertically by a factor of three and then by shifting two units to the left.

George and Laura are trying to graph 2sin(3x). Laura wants to do the horizontal compression followed by the vertical stretch, but George insists on the reverse order. Which method is correct? Why?

John wants to move up, and Teresa is willing to move to the left. Can they graph

sin〖(x+π/4)+1〗? What two graphing transformations would they use? Does the order of

the transformations matter?

State a general principle about transformations, on the basis of parts (a), (b), and (c).

3) Suppose that the x -intercepts of the graph of f are {4,7}. Find the x -intercepts of the graphs of the following functions. (If it is impossible to tell from the given information, explain why.)

(a) f (x − 3)

(b) 2 f (7x + 1)

(c) 7 f (x) + 1

4) Let A be the set of all humans, and let B be the set of all human males. Deﬁne f: A → B by the rule f (a) = a ’s biological father.

(a) In words, describe the range of f.

(b) If y ∈ B, then (in words) what is the set f -1 [{y}]?

(c) If x ∈ A, then (in words) what is the set f^(-1)[ f [{x}]]?

(d) If y ∈ B, then what is the set f [ f^(-1) [{y}]]. Your answer should depend on whether y is in the range of f.

5) Let f: A → B be a function.

(a) Prove that if S ⊆ A, then f^(-1) [ f [S]] ⊇ S.

(b) Prove that if T ⊆ B, then f [ f^(-1) [T]] ⊆ T.

6) Let f : R R, f(x) = x^2 (x-3).

Given a real number b, ﬁnd the number of elements in f^(-1) [{ b }]. (The answer will depend on b. It will be helpful to draw a rough graph of f, and you probably will need ideas from calculus to complete this exercise.)

Find three intervals whose union is R, such that f is injective as a function on each interval.

Use your answers to (a) and (b) to deﬁne the domain and codomain of three bijective function

7) Let F: R → R, F (x) = 3 x + 5. We can write F easily as the composite of two functions. Namely, if we deﬁne f and g (both with domain and codomain R ) by the rules f ( x ) = x + 5 and g ( x ) = 3 x , then F = f ◦ g . Find four other ways to write F as the composite of two functions (with domain and codomain R ).

Sample Solution

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