Practice Questions

If $(\frac{x}{2} + 3, y - 4) = (5, 1)$, calculate the values of $x$ and $y$.

If set A has 4 elements and set B = {x, y, z}, calculate the number of elements in A × B and the total number of possible relations from A to B.

Name the function $f: \mathbf{R} \rightarrow \mathbf{R}$ defined by $f(x) = k$, where $k$ is a real number.

Given the function $f(x) = \frac{2x+1}{x-3}$, calculate $f(5)$.

Let $A = \{1, 2, 3\}$ and $B = \{a, b\}$. Design two distinct functions, $f_1$ and $f_2$, from $A$ to $B$ such that the range of $f_1$ is $\{a\}$ and the range of $f_2$ is $\{a, b\}$. Justify why both are valid functions.

State the condition for two ordered pairs $(x_1, y_1)$ and $(x_2, y_2)$ to be equal.

If $(a+2, 9) = (5, b-1)$, find the values of $a$ and $b$.

If $P = \{a, b\}$ and $Q = \{1, 2, 3\}$, list the elements of $P \times Q$ and $Q \times P$. State whether $P \times Q$ is equal to $Q \times P$.

Define the Cartesian product of two non-empty sets A and B.

Examine if the relation $R = \{(2, 1), (4, 2), (6, 3), (8, 4), (6, 5)\}$ is a function. Provide a clear reason for your answer.

Let R be a relation on the set of natural numbers $\mathbf{N}$ defined by $R = \{(x, y) : y = 2x + 1, x \text{ is a natural number and } x \le 3\}$. List the elements of R and state its domain and range.

Let a relation R be defined from $A = \{2, 3, 4, 5\}$ to $B = \{3, 6, 7, 10\}$ by $R = \{(x, y) : x \text{ divides } y\}$. Write R in roster form. Also, find its domain and range.

Analyze the function $f(x) = \frac{x^2 + 2x + 1}{x^2 - x - 6}$. (i) Factorize the denominator. (ii) Use this to determine the domain of the function $f$. (iii) Calculate $f(0)$ and $f(-1)$.

Justify whether the relation $R$ on the set $\mathbb{Z}$ of integers, defined as $R = \{(a, b) : |a - b| \le 1\}$, is reflexive, symmetric, or transitive.

Describe the Identity Function and the Signum Function. For each function, state its domain, range, and provide its definition.

If $A = \{1, 2, 3\}$ and $B = \{a, b\}$, calculate $A \times B$ and $B \times A$.

Let R be a relation on the set of natural numbers $\mathbf{N}$ defined by $R = \{(x, y) : y = x^2, x \text{ is a prime number less than 10}\}$. Calculate the range of R.

Let $A = \{1, 2\}$, $B = \{3, 4\}$, and $C = \{4, 5\}$. Demonstrate that $A \times (B \cup C) = (A \times B) \cup (A \times C)$.

A relation R is defined on the set $A = \{1, 2, 3, 4, 5, 6\}$ by $R = \{(x, y) : y = x + 2\}$. Write R in roster form and determine its domain and range. The codomain is A.

If $f(x) = 2x - 5$ and $g(x) = x^2 + 1$ are two real functions, calculate $(f+g)(x)$, $(fg)(x)$, and $(\frac{f}{g})(x)$.

Explain the difference between the range and the codomain of a relation from a non-empty set A to a non-empty set B.

Identify which of the following relations are functions. Give a reason for your answer. (i) $R_1 = \{(a,1), (b,2), (c,1)\}$ where the domain is $\{a, b, c\}$. (ii) $R_2 = \{(x,5), (y,6), (x,7)\}$ where the domain is $\{x, y\}$.

If $A = \{-1, 0, 1\}$, list the elements of the set $A \times A$.

Define a function from a set A to a set B. Explain the two key conditions that a relation must satisfy to be a function. Provide one example of a relation that is a function and one that is not, explaining your reasoning.

Justify if the relation $R = \{(x, y) : x, y \in \mathbb{Z}, xy \ge 0\}$ is a function from $\mathbb{Z}$ to $\mathbb{Z}$.

Propose a real function $f(x)$ whose domain is $\mathbb{R} - \{-2, 2\}$ and whose range is $(0, \infty)$.

Evaluate whether the relation $f: \mathbb{N} \to \mathbb{N}$ defined by $f(n) = n + (-1)^n$ represents a valid function.

Given $A = \{x, y, z\}$ and $B = \{1, 2\}$. Let $R_1 = \{(x, 1), (y, 2), (z, 1)\}$ and $R_2 = \{(x, 1), (y, 1), (x, 2)\}$. Evaluate both $R_1$ and $R_2$ to determine if they can be considered functions from $A$ to $B$. Provide a detailed justification for each case.

Propose a function $f: \mathbb{Z} \to \mathbb{Z}$ such that its range is the set of all non-negative integers, $\mathbb{W} = \{0, 1, 2, ...\}$. Prove that your proposed relation is indeed a function with the specified range.

Design a function $f: \mathbb{R} \to \mathbb{R}$ and another function $g: \mathbb{R} \to \mathbb{R}$ such that $(f+g)(x)$ is defined for all $x \in \mathbb{R}$, but the domain of $(\frac{f}{g})(x)$ is $\mathbb{R} - \{k\}$ for some integer $k$. Define $f(x)$, $g(x)$, $(f+g)(x)$, and $(\frac{f}{g})(x)$, and explicitly state their domains.

Let $A = \{1, 2, 3, ..., 10\}$. Formulate a relation $R$ on $A$ defined by $R = \{(x, y) : y = 2x - 1\}$. (i) Express $R$ as a set of ordered pairs. (ii) Find the domain, codomain, and range of $R$. (iii) Critique whether $R$ qualifies as a function from $A$ to $A$. If not, modify the domain of $R$ to the largest possible subset of $A$ for which it would be a function. Justify your modifications.

If a set A has 3 elements and a set B has 4 elements, how many relations can be defined from set A to set B?

Calculate the domain and range of the real function $f(x) = \sqrt{25 - x^2}$.

Let $A = \{1, 2, 3, 4, 5\}$ and $R$ be the relation on $A$ defined by $R = \{(a, b) : b = a^2 - 1\}$. (i) Write $R$ in roster form. (ii) Find the domain, codomain, and range of $R$. (iii) Represent $R$ using an arrow diagram. (iv) Examine if $R$ is a function from $A$ to $A$. Justify your answer.

Let $f(x) = 3x - 2$ and $g(x) = x^2$ be two real functions. Define the following functions: (i) $(f+g)(x)$ (ii) $(f-g)(x)$ (iii) $(fg)(x)$ (iv) $(\frac{f}{g})(x)$

Prove that for any two non-empty sets A and B, $A \times B = B \times A$ if and only if $A = B$. Justify each step of your proof.

Let $f(x) = \frac{1}{\sqrt{x - |x|}}$. Critique the possibility of defining this function for any real number $x$. Determine its domain and justify your conclusion.

Let $A = \{1, 2, 3, 4, 5\}$. A relation R is defined on set A as $R = \{(a, b) : b = a+1, a,b \in A\}$. (i) Express R in roster form. (ii) State the domain of R. (iii) State the range of R. (iv) State the codomain of R. (v) Explain if R is a function on A.

Let $f(x) = x^2$ and $g(x) = \sqrt{x-1}$. Evaluate the domains of the functions $f+g$, $f-g$, $fg$, and $\frac{f}{g}$. Justify how the domain of the combined function is determined in each case.

Given $A \times B = \{(a, x), (b, y), (a, y), (b, x)\}$. (i) Identify the sets A and B. (ii) State the value of $n(A \times B)$. (iii) List all possible subsets of A. (iv) How many relations can be defined from set B to set A?

The Cartesian product $A \times A$ has 9 elements, among which are found $(1, 0)$ and $(2, 1)$. Find the set A and the remaining elements of $A \times A$.

A function $f: \mathbf{R} \rightarrow \mathbf{R}$ is defined by $f(x) = \begin{cases} 1 - x, & x < 1 \ 0, & x = 1 \ x - 1, & x > 1 \end{cases}$. (i) Calculate $f(-2)$. (ii) Calculate $f(1)$. (iii) Calculate $f(5)$. (iv) Analyze the function and determine its range.

Formulate a piecewise function $f(x)$ with three distinct linear pieces, such that its domain is $\mathbb{R}$ and its range is $[0, \infty)$. The function should not be the modulus function.

Formulate a relation on the set $A = \{1, 2, 3\}$ that is reflexive and symmetric but not transitive.

Critique the statement: "If $n(A) = m$ and $n(B) = n$, then the number of non-empty relations from A to B is $2^{mn} - 1$." Is this always true? Justify your answer.