Self-Assessment Quiz: Functions

Twenty questions to check your grip on functions, their properties, composition, and inverses. Answer before opening the key. Aim for 16+.

Question 1

A function $f\colon A \to B$ is, formally, a subset of $A \times B$ such that:

A) every $b \in B$ pairs with exactly one $a \in A$ B) every $a \in A$ pairs with exactly one $b \in B$ C) every $a \in A$ pairs with at least one $b \in B$ D) no two elements of $A$ pair with the same $b \in B$

Question 2

The phrase "exactly one output per input" encodes two conditions. They are:

A) injectivity and surjectivity B) totality (at least one) and single-valuedness (at most one) C) commutativity and associativity D) domain equals codomain, and range equals codomain

Question 3

For $f\colon \mathbb{R} \to \mathbb{R}$, $f(x) = x^2$, the codomain and the range are:

A) both $\mathbb{R}$ B) codomain $[0,\infty)$, range $\mathbb{R}$ C) codomain $\mathbb{R}$, range $[0,\infty)$ D) both $[0,\infty)$

Question 4

The notation $f^{-1}(T)$ for a set $T \subseteq B$ denotes:

A) the inverse function applied to $T$, which exists only if $f$ is bijective B) the set of all inputs whose output lies in $T$ — defined for every function C) the complement of $f(T)$ D) the image of $T$ under $f$

Question 5

A function $f$ is injective exactly when:

A) every codomain element is hit B) $f(a_1) = f(a_2)$ implies $a_1 = a_2$ (no collisions) C) the range equals the codomain D) $f$ has an inverse function

Question 6

A function $f\colon A \to B$ is surjective exactly when:

A) distinct inputs give distinct outputs B) $\operatorname{range}(f) = B$ (nothing is missed) C) $\lvert A \rvert = \lvert B \rvert$ D) $f$ is also injective

Question 7

In the composition $g \circ f$, the function applied to the input first is:

A) $g$ B) $f$ C) whichever is injective D) they are applied simultaneously

Question 8 (True/False, justify)

True or false: Function composition is commutative — $g \circ f = f \circ g$ for all $f, g$ for which both are defined. Justify in one sentence.

Question 9

The central theorem of the chapter states that a function $f\colon A \to B$ is invertible if and only if it is:

A) injective B) surjective C) bijective D) total

Question 10

In the proof that bijectivity implies invertibility, surjectivity supplies _ and injectivity supplies ___ of the inverse's value $g(b)$:

A) uniqueness; existence B) existence; uniqueness C) totality; surjectivity D) commutativity; associativity

Question 11

$\lfloor -3.4 \rfloor$ equals:

A) $-3$ B) $-4$ C) $3$ D) $4$

Question 12 (True/False, justify)

True or false: For all real $x$, $\lfloor x \rfloor$ equals $x$ with its decimal part chopped off ("truncation"). Justify in one sentence.

Question 13

Paginating $n = 47$ items at $10$ per page requires how many pages, and which function expresses it?

A) $\lfloor 47/10 \rfloor = 4$ B) $\lceil 47/10 \rceil = 5$ C) $47 \bmod 10 = 7$ D) $\lfloor 47/10 \rfloor = 5$

Question 14

Why must a hash function from all 64-bit integers into a 256-slot table have collisions?

A) hash functions are poorly designed B) the domain ($2^{64}$ values) is far larger than the codomain (256), so no injection exists C) modular arithmetic is not deterministic D) it actually need not have collisions if the hash is good

Question 15

Which property of a function makes lossless compression possible (so the file can always be recovered)?

A) surjectivity B) injectivity C) idempotence D) commutativity

Question 16

A pure function in programming corresponds to which mathematical idea?

A) a surjection B) a function in the §9.1 sense — output depends only on inputs (well-defined), with an output for every input (total) C) an inverse function D) the identity function

Question 17 (True/False, justify)

True or false: The function $x \mapsto x \bmod n$ from $\mathbb{Z}$ onto $\{0,1,\dots,n-1\}$ is surjective but not injective. Justify in one sentence.

Question 18

The mod, exponential, and log functions: which pair are inverse bijections of each other on their natural domains?

A) floor and ceiling B) $x \bmod n$ and $x \cdot n$ C) $b^{(\cdot)}\colon \mathbb{R} \to (0,\infty)$ and $\log_b\colon (0,\infty) \to \mathbb{R}$ D) $x^2$ and $\sqrt{x}$ on all of $\mathbb{R}$

Question 19 (Short answer)

State the two proof templates from §9.3 in one line each: how do you prove injectivity, and how do you prove surjectivity?

Question 20 (Short answer)

In one sentence, explain why the same rule $x \mapsto 2x$ can be a non-surjective function onto $\mathbb{Z}$ but a bijective function onto the even integers — i.e., why classification depends on more than the rule.

Answer Key

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