Problem sheet 1
Sets. Maxima and suprema. Functions: graph, expression from verbal description, composition
Problems from the book [CT]:
Canuto, Claudio, Tabacco, Anita, Mathematical Analysis I, Pearson 2022
E.1.1-a)-b); E.1.2-a)-b); E1.3; E.1.4.
E.2.1-a)-b) (Note that asking for the domain of the function means asking for the natural domain,
i.e. the largest subset of R in which the expression of f(x) makes sense).
E.2.2-a)-b)-d).
Problem 1
Represent on a Cartesian plane, where x represents the population of a country, and y its GNP (gross
national product) in $, the set of all countries whose population is above 1,000,000 inhabitants and
whose per capita GNP is between 1,000 and 10,000 $.
Problem 2
Let
A = [−1, 1) D1 = {x ∈ R :
x−2
∈ A}
x+2
D2 = {x ∈ R :
3x + 2
∈ A}
x−3
Find D1 ∪ D2 and D1 ∩ D2 .
Problem 3
For each of the following sets, find out their maximum and minimum (if they exist), their supremum
and infimum (that could be infinite). Justify the answers:
a) {x ∈ Q : 0 ≤ x, x2 ≤ 2}
b) {(−1)n
n
, n ∈ N}
n+1
c) {x ∈ R : x = 3n + (−2)n , n ∈ N}
Problem 4
(a) Find the domain and the analytic expression of the function that represents the upper half of
the circumference of equation (x − 1)2 + y 2 = 1.
(b) Find the domain and the analytic expression of the function that represents the lower part of
the parable of equation x + (y − 1)2 = 0.
(c) Find the domain and the analytic expression of the function whose graph is the segment joining
the points (2, −4), (4, −5).
Problem 5
In a city, surveys of the temperature from midnight to 2pm every two hours have produced the
following results:
t 0 2
4
6
8 10 12 14
T 23 26 29 32 33 33 32 32
Assuming that the function representing the temperature at any time t ∈ [0, 14] is piecewise linear,
joining the measured values,
• (a) Plot the graph of T (t);
• (b) Evaluate T (5).
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Problem 6
The owner of a furniture factory knows that to produce 100 chairs a day costs 2200 euros, while
producing 300 costs 4800 euros.
Express the cost as a function of the number of chairs, assuming the relation is linear.
Problem 7
An open box with a volume of 2m3 has a square base. Determine the area of the box as a function
of the base side length.
Problem 8
Find the domain and the expression for for the function whose graph consists of the line segment
between the points (−2, 2) and (−1, 0) together with the upper part of the circle with centre in the
origin and radius 1.
Problem 9
x+1
1
compute f ◦ g, g ◦ f, f ◦ f , making precise the
Given the functions f (x) = x + and g(x) =
x
x+2
domains in each case. [Notice that the functions f and g are assumed to be defined in their ‘natural’
domains, i.e. in the largest sets where the expressions make sense. Furthermore, when we consider
f ◦ g, we implicitly assume that the domain of g is restricted in such a way that Im(g) is contained
in the domain of f .]
Problem 10
If f and g are the functions defined by the following table .
x
f (x)
g(x)
1
3
6
2
1
3
3
4
2
4
2
1
5
2
2
6
5
3
evaluate (f ◦ g)(1), (g ◦ f )(1), (f ◦ f )(1), (g ◦ g)(1), (g ◦ f )(3), (f ◦ g)(6).
Problem 11
1+x
1
Given the function f (x) =
, compute f (−x), f ( x1 ), f ( 1−x
), f (f (x)).
1−x
Problem 12
After having plotted the graphs of the functions f (x) = x2 , f (x) = x, f (x) = x3 , obtain from them
the graphs of the following functions1 :
5) f (x) = −x2
6) f (x) = (x − 3)2
7) f (x) = −(x − 3)2
8) f (x) = 4 − (x − 3)2
9) f (x) = x2 − 6x + 4
10) f (x) = −x
11) f (x) = |x − 3|
1 do not worry that I start from 5)
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12) f (x) = −|x − 3|
13) f (x) = 2 − |x − 3|
14) f (x) = −x3
15) f (x) = −x3 + 1
16) f (x) = | − x3 + 1|
17) f (x) = |x3 | − 2
18) f (x) = −x2 + 4
19) f (x) = |4 − x2 |
20) f (x) = 2 − |4 − x2 |
21) f (x) = |2 − |4 − x2 ||
22) f (x) = −|x|
23) f (x) = 2 − |x|
24) f (x) = |2 − |x||
25) f (x) = −(|x| + 1)
26) f (x) = −(x − 1)3
27) f (x) = |(x − 1)3 |
28) f (x) = (x + 1)3 − 1
29) f (x) = −||x3 | − 2|
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