Multiple Correct Answers MCQs for Sub-Topics of Topic 9: Sets, Relations & Functions

Question 1. Which of the following are well-defined collections of objects?

(A) The collection of all months of a year.

(B) The collection of all intelligent students in your class.

(C) The collection of all even numbers less than $50$.

(D) The collection of all beautiful flowers in a garden.

Question 2. Let $A = \{x : x \text{ is an integer and } -2 \leq x < 3\}$. Which of the following elements belong to set $A$?

(A) $-2$

(B) $-1$

(C) $2$

(D) $3$

Question 3. Which of the following sets are represented in roster form?

(A) $\{a, b, c, d\}$

(B) $\{x : x \text{ is a vowel in the English alphabet}\}$

(C) $\{1, 2, 3, ...\}$

(D) $\{x^2 : x \in \mathbb{Z}, |x| \leq 2\}$

Question 4. Which of the following describe the set $\{2, 4, 6, 8\}$ using set-builder form?

(A) $\{x : x \text{ is an even natural number}\}$

(B) $\{x : x = 2n, n \in \mathbb{N}, 1 \leq n \leq 4\}$

(C) $\{x : x \text{ is an even number less than } 10\}$

(D) $\{x : x \text{ is an even integer between } 1 \text{ and } 9\}$

Question 5. Which of the following notations represent standard sets of numbers?

(A) $\mathbb{R}$ (Set of real numbers)

(B) $\mathbb{I}$ (Set of irrational numbers)

(C) $\mathbb{N}$ (Set of natural numbers)

(D) $\mathbb{C}$ (Set of complex numbers)

Question 6. Which of the following are true regarding ordered pairs?

(A) $(a, b) = (b, a)$ for all $a, b$.

(B) $(a, b) = (c, d)$ if and only if $a=c$ and $b=d$.

(C) The order of elements matters in an ordered pair.

(D) $(1, 2)$ is the same as the set $\{1, 2\}$.

Question 7. If $(2x, x+y) = (4, 5)$, which of the following statements are true?

(A) $2x = 4$

(B) $x+y = 5$

(C) $x=2$

(D) $y=3$

Question 8. Let $A = \{1, 2\}$ and $B = \{a, b\}$. Which of the following are elements of the Cartesian product $A \times B$?

(A) $(1, a)$

(B) $(b, 2)$

(C) $(2, b)$

(D) $\{1, b\}$

Question 9. If $n(A) = m$ and $n(B) = n$, then the number of elements in $A \times B$ is:

(A) $m+n$

(B) $m \times n$

(C) $n(A \cup B)$

(D) $n(B \times A)$

Question 10. Let $A = \{a, b\}$. Which of the following are elements of $A \times A$?

(A) $(a, a)$

(B) $(b, b)$

(C) $(a, b)$

(D) $\{a, b\}$

Question 1. Which of the following sets are empty sets?

(A) $\{x : x \text{ is an even prime number greater than } 2\}$

(B) $\{x : x \in \mathbb{N}, x^2 = 4\}$

(C) $\{x : x \in \mathbb{Z}, x^2 + 1 = 0\}$

(D) $\{x : x \text{ is a student in your class who is currently on the moon}\}$

Question 2. Which of the following sets are finite sets?

(A) The set of all people in India.

(B) The set of all natural numbers.

(C) The set of all positive integers less than $100$.

(D) The set of all stars in the sky.

Question 3. Which of the following sets are infinite sets?

(A) The set of points on the circumference of a circle.

(B) The set of prime numbers.

(C) The set of solutions to the equation $x^2 - 4 = 0$ in $\mathbb{R}$.

(D) The set of real numbers between $0$ and $1$ (exclusive).

Question 4. Which of the following are singleton sets?

(A) $\{0\}$

(B) $\phi$

(C) $\{x : x \in \mathbb{Z}, |x| = 1\}$

(D) $\{x : x \in \mathbb{R}, x^2 = 0\}$

Question 5. Let $A = \{1, 2, 3\}$. Which of the following sets are equal to $A$?

(A) $\{3, 1, 2\}$

(B) $\{x : x \in \mathbb{N}, x < 4\}$

(C) $\{1, 2, 3, 1\}$

(D) $\{x : x \in \mathbb{Z}, x^3 - 6x^2 + 11x - 6 = 0\}$

Question 6. Let $A = \{a, b, c\}$ and $B = \{1, 2, 3\}$. Which of the following statements are true?

(A) $A$ and $B$ are equal sets.

(B) $A$ and $B$ are equivalent sets.

(C) $n(A) = n(B)$

(D) $A \subseteq B$

Question 7. Let $A = \{1, 2\}$. Which of the following are subsets of $A$?

(A) $\{1\}$

(B) $\{1, 2\}$

(C) $\phi$

(D) $\{0\}$

Question 8. If $A$ is the universal set, which of the following statements are true?

(A) $A \subseteq A$

(B) $\phi \subseteq A$

(C) For any set $B$, $B \subseteq A$.

(D) $A$ has no proper subsets.

Question 9. Let $S = \{1, 2, 3\}$. Which of the following are elements of the power set $P(S)$?

(A) $\{1\}$

(B) $\phi$

(C) $\{1, 2, 3\}$

(D) $1$

Question 10. Which of the following intervals are subsets of the set of real numbers $\mathbb{R}$?

(A) $(0, 1)$

(B) $[-2, 5]$

(C) $(-\infty, \infty)$

(D) $\mathbb{N}$

Question 1. Let $A = \{1, 2, 3\}$ and $B = \{1, 2, 3, 4\}$. Which of the following statements correctly describe the relationship between $A$ and $B$?

(A) $A \subset B$

(B) $A \subseteq B$

(C) $B$ is a superset of $A$

(D) $A = B$

Question 2. A relation $R$ from set $A$ to set $B$ is defined as $R \subseteq A \times B$. If $A=\{1, 2\}$ and $B=\{p, q\}$, which of the following are possible relations from $A$ to $B$?

(A) $\phi$

(B) $\{(1, p)\}$

(C) $\{(1, p), (2, q)\}$

(D) $\{(1, q), (2, p), (1, p)\}$

Question 3. Let $A = \{1, 2, 3\}$. A relation $R$ on $A$ is defined as $R = \{(x, y) : x+y = 4\}$. Which of the following are elements of $R$?

(A) $(1, 3)$

(B) $(2, 2)$

(C) $(3, 1)$

(D) $(1, 2)$

Question 4. For the relation $R = \{(1, 3), (2, 2), (3, 1)\}$ on $A = \{1, 2, 3\}$, which of the following are correct?

(A) The domain of $R$ is $\{1, 2, 3\}$.

(B) The range of $R$ is $\{1, 2, 3\}$.

(C) $(2, 2)$ is in $R$.

(D) The codomain is $\{1, 2, 3\}$.

Question 5. Let $R$ be the relation "is a factor of" from set $A$ to set $B$, where $A=\{2, 3\}$ and $B=\{6, 9, 10\}$. Which of the following ordered pairs are in $R$?

(A) $(2, 6)$

(B) $(3, 9)$

(C) $(2, 10)$

(D) $(3, 10)$

Question 6. For the relation $R$ in Question 5, which of the following statements are true?

(A) The domain of $R$ is $\{2, 3\}$.

(B) The range of $R$ is $\{6, 9, 10\}$.

(C) The range of $R$ is $\{6, 9, 10\}$.

(D) The range of $R$ is $\{6, 9, 10\}$.

Question 7. If a relation $R$ from $A$ to $B$ is represented by an arrow diagram, the elements in $A$ from which arrows originate are part of the _______, and the elements in $B$ to which arrows point are part of the _______.

(A) Range, Domain

(B) Domain, Range

(C) Domain, Codomain

(D) Codomain, Range

Question 8. Let $A = \{1, 2\}$. Which of the following are relations on $A$?

(A) $\{(1, 1), (2, 2)\}$

(B) $\{(1, 2)\}$

(C) $\{(1, 1), (2, 2), (1, 2), (2, 1)\}$

(D) $\{(1, 3)\}$

Question 9. Let $R$ be the relation "is greater than" on the set $\{1, 2, 3\}$. Which of the following ordered pairs belong to $R$?

(A) $(2, 1)$

(B) $(3, 1)$

(C) $(3, 2)$

(D) $(1, 1)$

Question 10. If a relation $R$ from set $A$ to set $B$ contains all possible ordered pairs from $A$ to $B$, then $R$ is the:

(A) Empty relation

(B) Universal relation

(C) Identity relation

(D) Cartesian product $A \times B$

Question 1. Let $A$ be a non-empty set. Which of the following relations on $A$ are always reflexive?

(A) The identity relation $I_A = \{(a, a) : a \in A\}$

(B) The universal relation $R = A \times A$

(C) The empty relation $\phi$

(D) Any equivalence relation on $A$

Question 2. Let $A$ be a set of people. The relation $R$ on $A$ defined by $a R b$ if "a is a sibling of b" is:

(A) Reflexive

(B) Symmetric

(C) Transitive

(D) An equivalence relation

Question 3. Let $A$ be the set of all triangles in a plane. The relation $R$ on $A$ defined by $T_1 R T_2$ if "$T_1$ is similar to $T_2$" is:

(A) Reflexive

(B) Symmetric

(C) Transitive

(D) An equivalence relation

Question 4. Let $A = \{1, 2, 3\}$. Consider the relation $R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)\}$. Which properties does $R$ satisfy?

(A) Reflexive

(B) Symmetric

(C) Transitive

(D) Equivalence relation

Question 5. Let $R$ be a relation on $\mathbb{Z}$ defined by $a R b$ if $a+b$ is even. Which of the following properties does $R$ satisfy?

(A) Reflexive

(B) Symmetric

(C) Transitive

(D) Equivalence relation

Question 6. For the relation in Question 5, the equivalence class of $1$ is:

(A) The set of all even integers.

(B) The set of all odd integers.

(C) $\{1\}$

(D) $\{x \in \mathbb{Z} : x \text{ is odd}\}$

Question 7. Let $A$ be the set of all lines in a plane. The relation $R$ defined by $l_1 R l_2$ if "$l_1$ is perpendicular to $l_2$" is:

(A) Reflexive

(B) Symmetric

(C) Transitive

(D) An equivalence relation

Question 8. Let $A = \{1, 2, 3, 4\}$. The relation $R = \{(a, b) : a \leq b\}$ on $A$ is:

(A) Reflexive

(B) Symmetric

(C) Transitive

(D) An equivalence relation

Question 9. Which of the following statements are true about the empty relation $\phi$ on a non-empty set $A$?

(A) It is reflexive.

(B) It is symmetric.

(C) It is transitive.

(D) It is an equivalence relation.

Question 10. Which of the following statements are true about the universal relation $A \times A$ on a non-empty set $A$?

(A) It is reflexive.

(B) It is symmetric.

(C) It is transitive.

(D) It is an equivalence relation.

Question 1. Let $A = \{1, 2, 3\}$, $B = \{3, 4, 5\}$. Which of the following elements are in $A \cup B$?

(A) $1$

(B) $3$

(C) $4$

(D) $6$

Question 2. Let $A = \{1, 2, 3\}$, $B = \{3, 4, 5\}$. Which of the following elements are in $A \cap B$?

(A) $1$

(B) $3$

(C) $4$

(D) $\{3\}$

Question 3. Let $A = \{a, b, c\}$, $B = \{c, d, e\}$. Which of the following are true?

(A) $A \cup B = \{a, b, c, d, e\}$

(B) $A \cap B = \{c\}$

(C) $A - B = \{a, b\}$

(D) $B - A = \{d, e\}$

Question 4. Let $U$ be the universal set and $A$ be a set. Which of the following expressions are equal to $A'$ (the complement of $A$)?

(A) $U - A$

(B) $\{x \in U : x \notin A\}$

(C) The shaded region outside circle $A$ in a Venn diagram.

(D) $A \cap U$

Question 5. In a Venn diagram with two sets $A$ and $B$ within a universal set $U$, which regions are typically represented by circles?

(A) The universal set $U$

(B) Set $A$

(C) Set $B$

(D) The region $A \cap B$

Question 6. If $A$ and $B$ are disjoint sets, which of the following statements are true?

(A) $A \cap B = \phi$

(B) $A \cup B$ has $n(A) + n(B)$ elements if $A$ and $B$ are finite.

(C) $A - B = A$

(D) $B - A = B$

Question 7. For any set $A$, which of the following are true?

(A) $A \cup \phi = A$

(B) $A \cap \phi = \phi$

(C) $A \cup A = A$

(D) $A \cap A = A$

Question 8. Let $U$ be the universal set. Which of the following are true?

(A) $A \cup U = U$

(B) $A \cap U = A$

(C) $U' = \phi$

(D) $\phi' = U$

Question 9. Which of the following are equivalent to $(A \cup B)'$ by De Morgan's Laws?

(A) $A' \cap B'$

(B) $A' \cup B'$

(C) $(A')' \cup (B')'$

(D) $(A \cap B)''$

Question 10. Which of the following are equivalent to $(A \cap B)'$ by De Morgan's Laws?

(A) $A' \cup B'$

(B) $A' \cap B'$

(C) $(A \cup B)''$

(D) $(A')' \cap (B')'$

Question 1. For any sets $A, B, C$, which of the following identities are true?

(A) $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$

(B) $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$

(C) $A \cup B = B \cup A$

(D) $A \cup (B \cup C) = (A \cup B) \cup C$

Question 2. Which of the following are correct statements of De Morgan's Laws for sets $A$ and $B$?

(A) $(A \cup B)' = A' \cap B'$

(B) $(A \cap B)' = A' \cup B'$

(C) $A - B = A \cap B'$

(D) $B - A = B \cap A'$

Question 3. For two finite sets $A$ and $B$, which of the following expressions represent $n(A \cup B)$?

(A) $n(A) + n(B) - n(A \cap B)$

(B) $n(A - B) + n(B - A) + n(A \cap B)$

(C) $n(A) + n(B)$ if $A \cap B = \phi$

(D) $n(A) + n(B) - n(A \cup B)$

Question 4. If $n(A) = 25$, $n(B) = 30$, and $n(A \cup B) = 45$, which of the following are true?

(A) $n(A \cap B) = n(A) + n(B) - n(A \cup B)$

(B) $n(A \cap B) = 25 + 30 - 45 = 10$

(C) $n(A - B) = n(A) - n(A \cap B) = 25 - 10 = 15$

(D) $n(B - A) = n(B) - n(A \cap B) = 30 - 10 = 20$

Question 5. In a class of $50$ students, $30$ like Math and $25$ like Science. If $10$ students like both, how many students like at least one subject?

(A) $n(\text{Math} \cup \text{Science}) = n(\text{Math}) + n(\text{Science}) - n(\text{Math} \cap \text{Science})$

(B) The number of students who like at least one subject is $30 + 25 - 10 = 45$.

(C) The number of students who like only Math is $30 - 10 = 20$.

(D) The number of students who like only Science is $25 - 10 = 15$.

Question 6. Let $U$ be the universal set, and $A$ be a set. Which of the following are true?

(A) $n(A) + n(A') = n(U)$

(B) $n(\phi) = 0$

(C) $n(U') = 0$

(D) $n((A \cup A')') = 0$

Question 7. For three finite sets $A, B$, and $C$, which of the following are true?

(A) $n(A \cup B \cup C) = n(A)+n(B)+n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)$

(B) $n(\text{only A}) = n(A) - n(A \cap B) - n(A \cap C) + n(A \cap B \cap C)$

(C) $n(\text{only A and B}) = n(A \cap B) - n(A \cap B \cap C)$

(D) $n(\text{none of A, B, C}) = n(U) - n(A \cup B \cup C)$

Question 8. Let $A$ and $B$ be two finite sets. Which of the following are true?

(A) $n(A \Delta B) = n((A-B) \cup (B-A))$

(B) $n(A \Delta B) = n(A) + n(B) - 2n(A \cap B)$

(C) $n(A \Delta B) = n(A \cup B) - n(A \cap B)$

(D) $A \Delta B = (A \cup B) - (A \cap B)$

Question 9. In a survey, $60\%$ of people like product A, $50\%$ like product B, and $20\%$ like both. If everyone likes at least one product, which of the following are true?

(A) The percentage of people who like only A is $40\%$.

(B) The percentage of people who like only B is $30\%$.

(C) The percentage of people who like at least one product is $90\%$.

(D) The total percentage of people surveyed is $100\%$.

Question 10. Let $A$ and $B$ be finite sets. Which of the following is/are equivalent to $n(A \cap B)$?

(A) $n(A) + n(B) - n(A \cup B)$

(B) $n(A \cup B) - n(A - B) - n(B - A)$

(C) $n(A \cup B) - n(A) - n(B)$

(D) $n(A) - n(A - B)$

Question 1. Which of the following relations from $A=\{1, 2\}$ to $B=\{a, b\}$ are functions?

(A) $R_1 = \{(1, a), (2, b)\}$

(B) $R_2 = \{(1, a), (2, a)\}$

(C) $R_3 = \{(1, a), (1, b), (2, a)\}$

(D) $R_4 = \{(1, a)\}$

Question 2. If $f: A \to B$ is a function, which of the following statements are true?

(A) The domain of $f$ is $A$.

(B) The codomain of $f$ is $B$.

(C) For each $x \in A$, there is exactly one $y \in B$ such that $(x, y) \in f$.

(D) The range of $f$ is equal to the codomain $B$.

Question 3. Let $f = \{(1, p), (2, q), (3, r)\}$ be a function from $A=\{1, 2, 3\}$ to $B=\{p, q, r, s\}$. Which of the following statements are true?

(A) The domain of $f$ is $\{1, 2, 3\}$.

(B) The range of $f$ is $\{p, q, r\}$.

(C) The codomain of $f$ is $\{p, q, r, s\}$.

(D) $s$ is in the range of $f$.

Question 4. Which of the following sets represent the range of a function $f: A \to B$?

(A) The set of all second elements of ordered pairs in $f$.

(B) $\{f(x) : x \in A\}$

(C) A subset of the codomain $B$.

(D) The entire set $B$.

Question 5. If $f(x) = x^2$, with domain $\mathbb{R}$, which of the following are in the range of $f$?

(A) $4$

(B) $0$

(C) $-1$

(D) $9$

Question 6. If $f(x) = \sqrt{x}$, with domain $[0, \infty)$, which of the following are in the range of $f$?

(A) $2$

(B) $0$

(C) $-3$

(D) $5$

Question 7. Let $f: \mathbb{Z} \to \mathbb{Z}$ be defined by $f(x) = 2x$. Which of the following are in the range of $f$?

(A) $4$

(B) $0$

(C) $-6$

(D) $3$

Question 8. Which of the following relations from $A=\{a, b, c\}$ to $B=\{p, q\}$ are NOT functions?

(A) $\{(a, p), (b, q), (c, p)\}$

(B) $\{(a, p), (b, q)\}$

(C) $\{(a, p), (a, q), (b, p), (c, q)\}$

(D) $\{(a, p), (b, p), (c, q), (a, q)\}$

Question 9. If $f(x) = \frac{1}{x-1}$, with domain $\mathbb{R} - \{1\}$, which of the following values are in the range of $f$?

(A) $1$

(B) $-1$

(C) $0$

(D) $1/2$

Question 10. Let $f: \mathbb{N} \to \mathbb{Z}$ be defined by $f(n) = n - 5$. Which of the following are in the range of $f$?

(A) $-4$

(B) $0$

(C) $5$

(D) $-5$

Question 1. Which of the following functions from $\mathbb{R} \to \mathbb{R}$ are one-to-one (injective)?

(A) $f(x) = 2x + 5$

(B) $f(x) = x^2$

(C) $f(x) = x^3$

(D) $f(x) = |x|$

Question 2. Which of the following functions from $\mathbb{R} \to \mathbb{R}$ are onto (surjective)?

(A) $f(x) = 2x + 5$

(B) $f(x) = x^2$

(C) $f(x) = x^3$

(D) $f(x) = \sin x$

Question 3. Which of the following functions are bijective?

(A) $f: \mathbb{R} \to \mathbb{R}$, $f(x) = x^3$

(B) $f: \mathbb{Z} \to \mathbb{Z}$, $f(x) = x+1$

(C) $f: \mathbb{N} \to \mathbb{N}$, $f(x) = x^2$

(D) $f: \{1, 2\} \to \{a, b\}$, $f = \{(1, a), (2, b)\}$

Question 4. Which of the following functions from $\mathbb{R} \to \mathbb{R}$ are many-to-one?

(A) $f(x) = x^2$

(B) $f(x) = |x|$

(C) $f(x) = \cos x$

(D) $f(x) = 5$

Question 5. Which of the following functions from $\mathbb{R} \to \mathbb{R}$ are into (not onto)?

(A) $f(x) = x^2$

(B) $f(x) = |x|$

(C) $f(x) = \sin x$

(D) $f(x) = 2x + 5$

Question 6. Let $A = \{1, 2, 3\}$ and $B = \{p, q\}$. Which of the following functions from $A$ to $B$ are possible?

(A) One-to-one

(B) Onto

(C) Many-to-one

(D) Bijective

Question 7. Let $A = \{1, 2\}$ and $B = \{p, q, r\}$. Which of the following functions from $A$ to $B$ are possible?

(A) One-to-one

(B) Onto

(C) Into

(D) Many-to-one

Question 8. If $f: A \to B$ is an injective function, which of the following must be true?

(A) $n(A) \leq n(B)$

(B) $n(A) \geq n(B)$

(C) $f$ is also surjective.

(D) Different elements in $A$ have different images in $B$.

Question 9. If $f: A \to B$ is a surjective function, which of the following must be true?

(A) $n(A) \leq n(B)$

(B) $n(A) \geq n(B)$

(C) The range of $f$ is equal to $B$.

(D) Every element in $B$ has at least one pre-image in $A$.

Question 10. A function $f: A \to B$ is bijective if and only if:

(A) It is injective.

(B) It is surjective.

(C) For every $y \in B$, there is a unique $x \in A$ such that $f(x) = y$.

(D) It has an inverse function.

Question 1. A function $f: A \to B$ is a real function if:

(A) $A$ is a subset of $\mathbb{R}$.

(B) $B$ is a subset of $\mathbb{R}$.

(C) Both $A$ and $B$ are subsets of $\mathbb{R}$.

(D) The graph of the function can be drawn in the Cartesian plane.

Question 2. Which of the following are techniques used to find the domain of a real function?

(A) Identify values of the variable that make the denominator zero.

(B) Identify values of the variable that make an expression under an even root negative.

(C) Consider the context of the problem.

(D) Plot the graph and see the extent along the x-axis.

Question 3. The domain of the real function $f(x) = \frac{1}{x^2 - 4}$ is:

(A) $\mathbb{R} - \{2\}$

(B) $\mathbb{R} - \{-2\}$

(C) $\mathbb{R} - \{2, -2\}$

(D) $\mathbb{R} - \{4\}$

Question 4. The range of the real function $f(x) = x^2 + 1$ is:

(A) $\mathbb{R}$

(B) $[1, \infty)$

(C) $(-\infty, 1]$

(D) $\{y \in \mathbb{R} : y \geq 1\}$

Question 5. Which of the following statements are true about the graph of a real function $y = f(x)$?

(A) It is a visual representation of the set of ordered pairs $(x, f(x))$.

(B) Every vertical line intersects the graph at most once.

(C) The x-coordinates of the points on the graph form the domain.

(D) The y-coordinates of the points on the graph form the codomain.

Question 6. Which of the following are standard real functions?

(A) Identity function $f(x) = x$

(B) Constant function $f(x) = c$

(C) Modulus function $f(x) = |x|$

(D) Polynomial function $f(x) = a_n x^n + ... + a_0$

Question 7. Which of the following statements are true about the graph of the modulus function $f(x) = |x|$?

(A) It passes through the origin.

(B) It consists of two rays starting from the origin.

(C) It is symmetric about the y-axis.

(D) Its domain is $[0, \infty)$.

Question 8. The domain of the real function $f(x) = \frac{1}{\sqrt{9-x^2}}$ is:

(A) $(-3, 3)$

(B) $[-3, 3]$

(C) $\{x \in \mathbb{R} : 9 - x^2 > 0\}$

(D) $\{x \in \mathbb{R} : -3 < x < 3\}$

Question 9. Which of the following methods can be used to find the range of a real function $f(x)$?

(A) Analyzing the behavior of the function for different values in its domain.

(B) Solving $y = f(x)$ for x in terms of y and finding the values of y for which x is defined in the domain of f.

(C) Looking at the set of all y-coordinates covered by the graph of the function.

(D) Simply looking at the codomain of the function.

Question 10. Which of the following graphs represent functions?

(A) Graph 1

(B) Graph 2

(C) Graph 3

(D) Graph 4

Question 1. Let $f(x) = x+1$ and $g(x) = x^2$. Which of the following are correct?

(A) $(f+g)(x) = x+1+x^2$

(B) $(f-g)(x) = x+1-x^2$

(C) $(fg)(x) = (x+1)x^2$

(D) $(f/g)(x) = \frac{x+1}{x^2}$

Question 2. Let $f(x) = \sqrt{x}$ and $g(x) = x-2$. The domain of $f$ is $[0, \infty)$ and the domain of $g$ is $\mathbb{R}$. What is the domain of $(f+g)(x)$?

(A) $[0, \infty)$

(B) $\mathbb{R}$

(C) Intersection of domains of $f$ and $g$

(D) $\{x \in \mathbb{R} : x \geq 0\}$

Question 3. Let $f(x) = x^2$ and $g(x) = x-1$. The domain of $(f/g)(x)$ is the set of all real numbers $x$ such that:

(A) $x \neq 1$

(B) $g(x) \neq 0$

(C) $x \in$ Domain of $f$ and $x \in$ Domain of $g$ and $g(x) \neq 0$

(D) $x \in \mathbb{R} - \{1\}$

Question 4. Let $f(x) = x+2$ and $g(x) = x^2$. Which of the following are correct?

(A) $(f \circ g)(x) = (x+2)^2$

(B) $(f \circ g)(x) = x^2 + 2$

(C) $(g \circ f)(x) = (x+2)^2$

(D) $(g \circ f)(x) = x^2 + 2$

Question 5. The composition of functions $f \circ g$ is defined as $(f \circ g)(x) = f(g(x))$. Which condition(s) must be satisfied for $f \circ g$ to be defined for a specific $x$?

(A) $x$ must be in the domain of $g$.

(B) $g(x)$ must be in the domain of $f$.

(C) The range of $g$ must be a subset of the domain of $f$.

(D) The domain of $f$ must be a subset of the range of $g$.

Question 6. Let $f(x) = x^3$ and $g(x) = 1/x$. Find $(f \circ g)(x)$ and its domain.

(A) $(f \circ g)(x) = (1/x)^3 = 1/x^3$

(B) $(f \circ g)(x) = x^3$

(C) The domain of $(f \circ g)(x)$ is $\mathbb{R}$.

(D) The domain of $(f \circ g)(x)$ is $\mathbb{R} - \{0\}$.

Question 7. Composition of functions is associative. Which of the following represents the associative property?

(A) $f \circ g = g \circ f$

(B) $f \circ (g \circ h) = (f \circ g) \circ h$

(C) $(f \circ g)(x) = (g \circ f)(x)$

(D) $f(g(h(x))) = (f(g))(h(x))$

Question 8. Let $f(x) = 2x$ and $g(x) = x+3$. Which of the following are correct?

(A) $(f+g)(x) = 3x+3$

(B) $(fg)(x) = 2x(x+3) = 2x^2 + 6x$

(C) $(f \circ g)(x) = 2(x+3) = 2x + 6$

(D) $(g \circ f)(x) = 2x + 3$

Question 9. If the domain of $f$ is $D_f$ and the domain of $g$ is $D_g$, then the domain of $(f+g)$ is:

(A) $D_f \cup D_g$

(B) $D_f \cap D_g$

(C) $\{x : x \in D_f \text{ and } x \in D_g \}$

(D) $\{x \in D_f \cup D_g : f(x)+g(x) \text{ is defined} \}$

Question 10. Let $f(x) = \sqrt{x}$ and $g(x) = \sin x$. The domain of $f$ is $[0, \infty)$ and the domain of $g$ is $\mathbb{R}$. What is the domain of $(f \circ g)(x) = f(g(x)) = \sqrt{\sin x}$?

(A) $\{x \in \mathbb{R} : \sin x \geq 0\}$

(B) $\{x \in \mathbb{R} : x \in [2n\pi, (2n+1)\pi], n \in \mathbb{Z}\}$

(C) $[0, \infty)$

(D) $\mathbb{R}$

Question 1. A function $f: X \to Y$ is invertible if and only if it satisfies which of the following conditions?

(A) $f$ is one-to-one.

(B) $f$ is onto.

(C) $f$ is bijective.

(D) There exists a function $g: Y \to X$ such that $f(g(y)) = y$ for all $y \in Y$ and $g(f(x)) = x$ for all $x \in X$.

Question 2. If $f: X \to Y$ is an invertible function, its inverse $f^{-1}: Y \to X$ has which of the following properties?

(A) It is unique.

(B) Its domain is the codomain of $f$.

(C) Its range is the domain of $f$.

(D) $f \circ f^{-1} = I_Y$ and $f^{-1} \circ f = I_X$

Question 3. Which of the following real functions are invertible over their natural domains and codomains $\mathbb{R}$?

(A) $f(x) = x + 5$

(B) $f(x) = x^2$

(C) $f(x) = \frac{1}{x}$

(D) $f(x) = x^3$

Question 4. If $f(x) = ax+b$ where $a \neq 0$, which of the following are true about its inverse?

(A) The inverse exists.

(B) The inverse is $f^{-1}(x) = \frac{x-b}{a}$.

(C) The inverse is also a linear function.

(D) The graph of $f^{-1}(x)$ is the reflection of the graph of $f(x)$ across the line $y=x$.

Question 5. A binary operation $*$ on a set $S$ has which of the following properties?

(A) It is a function from $S \times S$ to $S$.

(B) For any $a, b \in S$, $a * b$ is a unique element in $S$.

(C) The operation must be commutative.

(D) The operation must have an identity element.

Question 6. Consider the set of integers $\mathbb{Z}$. Which of the following are binary operations on $\mathbb{Z}$?

(A) Addition ($+$)

(B) Subtraction ($-$)

(C) Multiplication ($\times$)

(D) Division ($\div$)

Question 7. Which of the following binary operations on $\mathbb{Z}$ are commutative?

(A) $a * b = a + b$

(B) $a * b = a - b$

(C) $a * b = a \times b$

(D) $a * b = \max(a, b)$

Question 8. Which of the following binary operations on $\mathbb{Z}$ are associative?

(A) $a * b = a + b$

(B) $a * b = a - b$

(C) $a * b = a \times b$

(D) $a * b = a^b$

Question 9. For the set of integers $\mathbb{Z}$ with the operation of addition ($+$), which of the following are true?

(A) $0$ is the identity element.

(B) The inverse of an element $a$ is $-a$.

(C) Every element in $\mathbb{Z}$ has an inverse under addition.

(D) $\mathbb{Z}$ is closed under addition.

Question 10. For the set of non-zero real numbers $\mathbb{R}^*$ with the operation of multiplication ($\times$), which of the following are true?

(A) $1$ is the identity element.

(B) The inverse of an element $a$ is $1/a$.

(C) Every element in $\mathbb{R}^*$ has an inverse under multiplication.

(D) $\mathbb{R}^*$ is closed under multiplication.