Mathematics/MATHS 108

Mathematics/MATHS 108

Show all working; problems that do not show their work will typically receive reduced or zero
marks. Late assignments cannot be marked under any circumstances.
1. (Sets.) Let A = (??3; 3);B = [??4; 2];C = [??1; 0) and D = f??4;??5g.
(a) Write the set A \ B as an interval. Is this an open interval? Is it a closed interval?
(b) Write the set (B n D) [ A as an interval. Is this an open interval? Is it a closed interval?
(c) Write the set B [ C as an interval. Is this an open interval? Is it a closed interval?
2. (Functions and relations.)
(a) Consider the relation
R = f(x; y) : x; y are any two people enrolled in Maths 108 where x is taller than y:g
Is R a function? If it is, explain why, and give its domain; if it is not, explain why not.
(b) Consider the relation
S = f(x; y) : x is any person enrolled in Maths 108, and y is the rst digit of their UoA ID number.g
Is S a function? If it is, explain why, and give its domain; if it is not, explain why not.
3. (Domain and range.) Here are three functions:
(a) f(x) =
x3 ?? 3×2 ?? 4x + 12
x2 ?? 4
; (b) g(x) = ln(16 ?? x2); (c) h(x) = e??x2
:
For each function, describe its natural domain and corresponding range. Explain your reasoning.
4. (Graphing.) Here are two more functions:
(a) f(x) =
8<
:
1=x; x 2 (??1; 0);
sin(x); x 2 [0; );
cos(x); x 2 [;1):
(b) g(x) =
x2 + 1
x2 ?? 2
:
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Draw the graphs of each of these functions by hand. Label any horizontal and vertical asymptotes
and any points of discontinuity. Show the work you used to nd these pieces of information.
5. (Continuity.) Here are two more functions:
(a) f(x) =

cx + 2; x 2 (??1; );
cos(x); x 2 [;1):
(b) h(x) =

x2 ?? 3c2; x 2 (??1; 1];
(cx)2; x 2 (1;1):
For each of these functions, nd a value of c such that the function is continuous everywhere. Explain
how you chose your values of c.
6. (Limits.) Calculate the following four limits, or show that they do not exist. Show your work.
(a) lim
x!0
ln(jx2 ?? 3j) (b) lim
x!1
9×3 ?? 3x
7 ?? 3x + 4×4 (c) lim
x!0
1 ??
p
1 ?? sin2(x)
x
(d) lim
x!1
e2x ?? ex
e2x + ex
7. (Counterexamples.)
The two statements written in the list below are false. For each statement listed below, come
up with a counterexample: that is, nd a function f(x) for statement (a) and a pair of functions
g(x); h(x) for statement (b) that demonstrate why the given statement is false. Explain why your
functions are counterexamples to the given statements.
(a) If a function f(x) is not continuous everywhere (that is, f(x) is a function with at least one
point of discontinuity,) then its square (f(x))2 is also not continuous everywhere.
(b) If the functions g(x) and h(x) do not have a limit as x goes to 0, then their ratio
g(x)
h(x)
also
does not have a limit as x goes to 0.
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