Questions

Q1. For each of the following functions, determine whether it is injective, surjective, or bijective; moreover, in case it is bijective, find its inverse.

(i) f : R → R defined by f (x) = x⁴.

(ii) f : R → [0, ∞) defined by f (x) = x⁴.

(iii) f : (−∞, 0] → [0, ∞) defined by f (x) = x⁴.

(iv) f : R → R defined by f (x) = x³ + 1.

Q2. For each of the following pairs of sets, determine whether any of the relations =,

⊆, ⊇ holds between them. Justify your answers.

(i) N and A = {x ∈ R : 1/x ∈ Q}.

(ii) B = {x ∈ R : 3x ∈ Q} and Q.

(iii) C = {x ∈ R : 4x ∈ Z} and D = {x ∈ R : x − 3 ∈ Z}.

(iv) Q and E = {x ∈ R : 1/(2 − x) ∈ Q}.

Q3. Write each of the following subsets of R as a union of one or more intervals. Use as few intervals as possible in each case. (Here a singleton a where a R is considered to be an interval.)

(i) [2, 5] ∩ (3, 9].

(ii) (−∞, 5) ∪ (3, 7].

(iii) (−4, 1) ∪ (−1, 2) ∪ {π}.

(iv) {x ∈ R : x² > 144} ∪ {12}.

(SUM) Q4. Let f (x) = ¹ and g(x) = e^−x. Find the following expressions:

(i) f (2 + x).

(ii) f (2x).

(iii) f (x²).

(iv) f ◦ f (x).

(v) f ( ¹ ).

(vi) f ◦ g(x).

Q5. Use sign analysis to solve the inequality

x³(x 3)²(x + 4)

x2 − 1 ≥ 0

and write the set of its solutions by using interval notation.

Q6. (i) Let f : R → R be the function defined by the rule f (x) = x² −x− 1. Evaluate

f at:

−1, (1 +

5)/2, π, x + 1, 3t + y.

Give your answers exactly, not as decimals (so you might involve the symbol

π in your answer, for example).

(ii) The function f : R R is defined by the rule f : x 9^x, for all x R.

(a) Find the outputs of f corresponding to the inputs:

−3, −5/2, −1/2, 1/2, 3/2.

(b) What input has 81 as its output?

(c) Are 0 and 1 outputs of f ?

(d) What is the image of f ?

(iii) Determine whether the rule

“associate to the number x the number y such that y² = x + 1” defines a function:

(a) from R to R;

(b) from [−1, ∞) to R;

(c) from [−1, ∞) to [0, ∞).

Q7. Use the Domain Convention to determine the domain and the range of the real valued functions of a real variable defined by the following rules:

(a) f (x) = 2x² + 1;

(b) g(x) = (2 − x)/(3 + x).

Is either of these functions one-to-one? If so, find an appropriate real-valued inverse

function.

Q8. Let f and g be both elementary functions. Show the following functions

M (x) = max{f (x), g(x)};

m(x) = min{f (x), g(x)},

are also elementary functions.

Q9. For each of the following functions, determine its infimum and supremum and whether it has a (global) maximum and/or a minimum.

(a) f (x) = 3^x;

(b) g(x) = 1/(1 + x²);

(c) h(x) = sin(2x).

Q10. Prove the following statements by using the definition of limit.

(i) lim 1 = 0.

x→∞ x2 + x

(ii) lim  1  = .

x→0 x4

(SUM) Q11. (i) Determine whether the following functions are even or odd (or neither): (a) g(x) = x · 2x−1 ,

(b) h(x) = x + sin x,

(c) k(x) = x³ + cos(πx).

(ii) Let a, b  R. Let f : R  R given by f (x) = a sin x + b cos x. What can you say about a and b if f is an odd function? What can you say if f is an even function? Is it possible for f to be both even and odd?

Q12. Prove the following limit by using the definition of limit.

(i) lim x 2 = ∞.

(ii) x→∞√  .

lim

x→1

3 x = 1

Extra  Questions

[Questions marked with a * may be more challenging than others.]

EQ1. If S is a finite set, we write S for the number of elements of S. The nonnegative integer S is also called the cardinality of S.

(i) Compute |{200, 2, √2}|, |{fish, pear}|, and |{200, 2, √2, 200}|.

(ii) Let A and B be finite disjoint sets, i.e. sets such that A ∩ B = ∅. Express

|A ∪ B| in terms of |A| and |B|.

(iii) Let A = {0, 1, 2, 3} and B = {2, 5, 6, 7, 8, 9}. What is |A ∪ B|? Compare it with |A| + |B|.

(iv) Let A and B be finite sets. Find a formula for |A ∪ B| in terms of |A|, |B|

and |A ∩ B|. Explain why your formula works.

EQ2. Give examples of:

(i) a function whose domain is not equal to the codomain;

(ii) a function whose domain is not equal to the image;

(iii) a function whose codomain is not equal to the image;

(iv) a function f : R → R such that f ◦ f = f ;

(v) two functions f, g : [0, ∞) → [0, ∞) such that f ◦ g and g ◦ f are not equal;

(vi) two different functions f, g : R → R whose restrictions to [−1, 0] are equal.

EQ3. (i) By expanding the expression (a − b)², or otherwise, prove that

a² + b²

ab ≤ 2

for all real numbers a and b.

(ii) Deduce further that

abcd ≤

a⁴ + b⁴ + c⁴ + d⁴

4

for all real numbers a, b, c and d.

(iii) Is it true that

abc ≤

a³ + b³ + c³

3

for all real numbers a, b and c? Justify your answer.

EQ4. Recall the Triangle Inequality for real numbers: If a, b ∈ R then

|a + b| ≤ |a| + |b|.

(i) Using the Triangle Inequality prove that if a, b ∈ R then

|a| − |b| ≤ |a − b|.

(ii) Deduce further that if a, b ∈ R then

||a| − |b|| ≤ |a − b|.

(iii) For which real numbers a, b does this last inequality hold with equality?