1.4. FUNCTIONS 39
Problem 1.93 Which of the following sets are functions? Part (a) requires you be very careful
with the definition of a function.
1. e set of points .cos.t/; 20 cos.t// where 1 < t < 1
2. All pairs .x; y/ where x
2
D y
3
3. e points (1,2), (0,4), (-1,2), (2,5), and (-2,4)
4. e points (2,1), (4,0), (2,-1), (5,2), and (4,-2)
5. e set of points .x; y/ where
x
2
C y
2
D 4
and y 0
6. e set of points .x; y/ where
x
2
C .y 1/
2
D 4
and y 0
Problem 1.94 For each of the functions in Problems 1.91 and 1.92 say if the functions are odd,
even, or neither.
Problem 1.95 Describe carefully the largest subset(s) of a circle that are functions.
Problem 1.96 If f .x/ is an odd function show that g.x/ D x f .x/ is an even function.
Problem 1.97 If h.x/ is an even function show that r.x/ D x h.x/ is an odd function.
Problem 1.98 Construct an infinite set of points that has the property that the largest function
that is a subset of the set contains one point.
Problem 1.99 Prove by logical argumentation that any subset of a function is a function. Re-
member that the definition of a function is that a function is a set of points with a special
property; your argument should not touch on the graph of the function at all.
Problem 1.100 Can the graph of a function enclose a finite area? Explain your answer.
40 1. REVIEW OF ALGEBRA
Problem 1.101 Using any method, compute the domain and range of the function
f .x/ D
ax C b
cx C d
where a; b; c, and d are real numbers. You may assume that the top and bottom of the fraction
do not cancel out so as to remove the variables and that a; b ¤ 0.
Problem 1.102 Suppose that f .x/ and g.x/ are both functions whose domain is all real num-
bers: .1; 1/. If we define h.x/ to be equal to f .x/ when x is negative and to equal g.x/ when
x is positive or zero, is h.x/ a function? Explain.
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