Relations and Functions

If $A = \{1, 2, 3\}$ and $B = \{x, y\}$, find the total number of elements in the Cartesian product $A \times B$. The following diagram represents the mapping of elements from set $A$ to set $B$ forming the product pairs.

If $A = \{7, 8\}$ and $B = \{5, 4, 2\}$, which element is in $A \times B$?

If $n(A) = 3$ and $n(B) = 4$, how many elements are in $(A \times B) \cap (B \times A)$ if $A \cap B = \emptyset$?

Find $y$ if $(x/3 + 1, y - 2/3) = (5/3, 1/3)$.

If the Cartesian product $A \times A$ has $9$ elements, among which are found $(-1, 0)$ and $(0, 1)$, find the set $A$. The figure illustrates the positions of these elements on a grid where the horizontal and vertical axes both represent set $A$.

Let $A$ and $B$ be two sets such that $n(A) = 3$ and $n(B) = 2$. If $(x, 1), (y, 2), (z, 1)$ are elements of $A \times B$, where $x, y, z$ are distinct elements, determine the set $A$. The diagram shows the points representing these elements in a coordinate-like plane.

If $n(A \times B) = 6$ and three elements of $A \times B$ are $(1, 3), (2, 5), (3, 3)$, then $A \times B$ is:

A computer program generates pairs $(x, y)$ from $A = \{1, 2\}$ and $B = \{1, 2\}$. It creates a new set $C = \{(x+y, x-y) : (x, y) \in A \times B\}$. What is the sum of all first coordinates in set $C$?

If $A \times B = \{(a,1), (b,3), (a,3), (b,1), (a,2), (b,2)\}$, identify the set $B$ based on the vertical projection shown in the diagram.

Let $A = \{1, 2, 3\}$. The set $A \times A$ is plotted on a grid. A line segment is drawn between $(1,1)$ and $(3,3)$. How many elements of the Cartesian product $A \times A$ lie on this line segment?

What is the inverse relation $R^{-1}$ if $R = \{(1, 2), (3, 4), (5, 6)\}$?

In the arrow diagram shown, which element in the domain has no image?

The identity relation $I_A$ on set $A = \{a, b, c\}$ is defined as:

Let $R$ be a relation on $\mathbb{N}$ defined by $xRy$ if $x + 2y = 8$. The domain of $R$ is:

In the set $A = \{1, 2, 3, 4\}$, let $R = \{(1,2), (2,3), (1,3)\}$. This relation is:

Consider the set $A = \{1, 2, 3\}$. The relation $R = \{(1,1), (2,2), (1,2), (2,1)\}$ is:

Let $R$ be a relation on the set $N$ of natural numbers defined by $nRm$ if $n$ is a factor of $m$. The relation is:

A set of heights $H = \{150, 160, 170\}$ is used to define a relation $R$ on $H$ such that $h_1 R h_2$ if $h_1 \geq h_2$. Find the number of elements in $R$.

In a manufacturing process, a relation $R$ is defined on the set $A = \{2, 3, 4, 5, 6\}$ such that $a R b$ if $|a - b|$ is a prime number. How many elements of the form $(a, a)$ exist in $R$?

A digital circuit uses a set $S = \{0, 1, 2, 3\}$. A relation $R$ on $S$ is given by $R = \{(x, y) : x^2 + y^2 \leq 4\}$. How many elements are in this relation?

Given the function $f(x) = x^2$ as shown in the graph below, identify the domain and range of the function when $x$ is defined for all real numbers.

Identify the function $f(x)$ whose graph is shown as a horizontal line passing through the origin.

The graph represents $f(x) = x^3$. What is the value of $f(-1)$?

Which set represents the domain of the function shown, given the graph exists only between the marked points?

The graph of the Greatest Integer Function $f(x) = [x]$ is shown. What is the value of $f(2.7)$?

Consider $f(x) = x + 2$ and $g(x) = x^2$. The diagram shows $g(f(x))$. What is the value of $g(f(1))$?

The graph of $f(x) = \frac{1}{x^2}$ is shown. What happens to $f(x)$ as $x$ approaches $0$ from the right?

A projectile's vertical velocity is $v(t) = |10 - 2t|$. At what times $t$ is the velocity exactly $4$ m/s?

A thermal expansion profile is given by $f(x) = [x] + \sqrt{x - [x]}$. If the input temperature is $x = 2.25$, what is the calculated expansion value?

A data filter uses the function $f(x) = \frac{1}{1 + e^{-x}}$. As the input $x$ becomes extremely large, what value does the output of the filter approach?

If $f(x) = c$ is a constant function and $f(2) = 5$, what is the value of $f(10)$?

Find the range of the function $f(x) = |x-1|$ as shown in the graph.

What is the range of the Signum function?

What is the value of $[ -1.2 ]$ where $[.]$ denotes the greatest integer function?

Given $f(x) = [x]$, what is the value of $f(0.9) + f(-0.1)$?

Which of these is the graph of a polynomial function of degree 2 with no real roots?

The graph shown represents $f(x) = |x|/x$ for $x \neq 0$. What is the value of $f(f(-2))$?

A digital artist is designing a repeating architectural pattern based on the function $f(x) = |x| + [x]$, where $[x]$ denotes the greatest integer function. During the design process, the artist needs to calculate the average rate of change of the function over a specific interval. If the artist analyzes the behavior of this function on the interval $x \in [1.2, 2.8]$, what is the value of the function at the midpoint of this interval?

A light beam follows the path $y = \frac{1}{x}$ for $x > 0$. If a mirror is placed at $y = x$, at what coordinate $(x, y)$ will the light beam reflect if it intersects the mirror's path?

Consider the polynomial function $P(x) = x^3 - x$. At which of the following sets of points does the graph intersect the x-axis?

If $f(x) = x^2 + 1$ and $g(x) = 3x - 2$, find $(f+g)(-1)$.

If $f(x) = \sqrt{x}$ and $g(x) = x$, find the value of $(g - f)(4)$.

Given $f(x) = x + 2$ and $g(x) = x - 2$, find the value of $(f \times g)(x)$ when $x = 5$.

Let $f(x) = 2$ be a constant function and $g(x) = x^2$. Find $(f \times g)(3)$.

Let $f$ and $g$ be two real functions defined by $f(x) = \sqrt{x + 2}$ and $g(x) = \sqrt{4 - x^2}$. The graph of $f(x)$ is shown below. Find the domain of the function $(f + g)(x)$.

Given two real-valued functions $f(x) = \sqrt{x+2}$ and $g(x) = \sqrt{4-x^2}$, let $h(x) = (f + g)(x)$. The graph of the region where both functions are simultaneously defined is shown below on the x-axis. Find the domain of the function $h(x)$.

If $f(x) = x^2$ and $g(x) = \sin(x)$, what is the domain of the function $(f/g)(x)$ on the interval $(0, \pi)$?

A logistics company calculates the cost of shipping $x$ metric tons of cargo using two components: a base handling fee $c(x) = x^2 + 5x$ and a fuel surcharge $s(x) = 2x + 10$. A client is offered a special discount where the final billable rate is determined by the quotient function $q(x) = \left(\frac{c}{s}\right)(x)$. If the client ships $x = 5$ metric tons, and the company applies an additional service tax of 10% on the value of $q(5)$, what is the final amount the client pays?

A chemical reaction in a lab is modeled by two functions. The concentration of chemical A over time $t$ is given by $f(t) = \sqrt{t + 4}$, and the concentration of chemical B is given by $g(t) = \sqrt{9 - t^2}$, where $t$ is in minutes. To study the interaction effect, a researcher defines a new function $h(t) = (f + g)(t)$. What is the domain of this combined real function $h(t)$ representing the time interval during which both chemicals are present?

An environmental agency monitors noise levels $N_1(x) = \ln(x)$ and $N_2(x) = \ln(2x)$ where $x$ is the distance from a source. They define a total interference function $I(x) = (N_1 + N_2)(x)$. Express $I(x)$ as a single logarithmic function.