Relations and Functions

Given the set $A = \{1, 2, 3\}$ and $B = \{4, 5, 6\}$, which of the following diagrams correctly represents a relation from set $A$ to set $B$ defined by $R = \{(1, 4), (2, 5), (3, 6)\}$?

What type of relation is shown in the mapping from set $P$ to set $Q$?

In the given diagram, which value of $x$ makes $f(x) = 0$ for the function $f(x) = x - 2$?

If a relation is defined as $y = -x$, which of the following lines represents it?

A relation $R$ is defined on the set of integers $\mathbb{Z}$ by the rule $y = x^2 - 1$. The graph of this relation is shown below for a continuous domain to help visualize the curve. If the domain is restricted to the set $A = \{-2, -1, 0, 1, 2\}$, which of the following represents the range of this relation?

Consider the mapping of a square function $g(x) = x^2$ for the domain $\{ -2, -1, 0, 1, 2 \}$. How many distinct elements are in the range of this function?

A function is defined by the rule $f(x) = x^2 - 1$. Based on the parabolic curve shown in the diagram, what are the coordinates of the points where the function intersects the x-axis?

A logistics company uses a cost function $C(x)$ to calculate the delivery charge for a package based on its weight $x$ (in kg). The function is defined as $C(x) = 50 + 20x$ for $0 < x \le 5$ and $C(x) = 150 + 10(x - 5)$ for $x > 5$. The graph shows the relationship between weight and cost. If a customer paid Rs 240 for a delivery, determine the weight of the package and identify whether the relation representing the total daily revenue $R(n)$ from $n$ packages of fixed weight is a linear function.

A function $f(x)$ is represented by the semi-circle shown in the diagram with radius 5 centered at the origin (upper half only). What is the algebraic expression for this function?

The identity function $f(x) = x$ and the constant function $g(x) = 5$ intersect at a specific point. What are the coordinates of this intersection point?

Consider the mapping of elements from set $A$ to set $B$ as shown in the diagram. If this relation is represented as a set of ordered pairs $(x, y)$, which of the following represents the relation?

Two ordered pairs $(x + y, 1)$ and $(5, x - y)$ are equal. Find the values of $x$ and $y$.

If the ordered pair $(2a - 5, a + 1)$ is equal to $(1, 4)$, find the value of $a$.

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

A point $P$ lies on the curve $y = x^2 - 2$. If the ordered pair representing the point is $(k, 7)$, find the possible positive value of $k$.

Given $(x^3, y^2) = (27, 16)$. Find all possible ordered pairs for $(x, y)$.

If $(a+2b, a-2b) = (10, 6)$, find the ordered pair $(a, b)$.

A logistic company tracks the movement of its delivery trucks on a Cartesian plane where the position of a truck at time $t$ is represented by the ordered pair $(x, y)$. For a specific truck, the coordinates satisfy the following system of equations involving the variables $a$ and $b$: $(a + 2b, 2a - b) = (7, 4)$. If the total distance traveled by the truck from the origin $(0,0)$ to the point $(a, b)$ follows a straight line path, what is the distance $(a, b)$ from the origin?

Two points in a geometry software are $A = (3a - b, 2a + b)$ and $B = (10, 15)$. If $A$ and $B$ are the same point, what is the value of $a + b$?

A chemistry experiment tracks the pressure $P$ and volume $V$ as an ordered pair $(P + V, PV)$. If the observed data for a gas is $(15, 50)$ and we know $P > V$, find the values of $P$ and $V$.

If the set $A$ has 3 elements and the set $B = \{3, 4, 5\}$, then the number of elements in $A \times B$ is:

The Cartesian product $P \times P$ has 16 elements. How many elements are in $P$?

If $A = \{1, 2\}$, the number of elements in $A \times A \times A$ is:

Find the value of $x$ and $y$ if $(x + 2, 4) = (5, y - 1)$.

Consider the set $S = \{x, y\}$ where $x$ and $y$ are the coordinates of points shown in the graph. If $A = \{1, 2\}$ and $B = \{1, 3\}$, and we define a subset of $A \times B$ as $R = \{(a, b) : a < b\}$. Which of the following correctly identifies the number of elements in $R$?

Let $A$ and $B$ be two sets such that $n(A) = 3$ and $n(B) = 4$. If three elements of the Cartesian product $A \times B$ are $(-1, a)$, $(0, b)$, and $(1, c)$, where $a, b, c \in B$ are distinct, and the diagram shows the mapping of these points on a plane where $A = \{-1, 0, 1\}$. How many total elements are in the set $A \times B$?

Given the sets $A = \{1, 2\}$ and $B = \{3, 4, 5\}$, let $R$ be the Cartesian product $A \times B$. The elements of $A \times B$ are represented as points $(x, y)$ in the coordinate plane. If we visualize this product on a grid, what is the area of the smallest rectangle that encloses all the points in $A \times B$?

A research team is mapping the potential habitats of two species in a coordinate plane. Species A is confined to the region defined by set $P = \{x \in \mathbb{Z} \mid 1 \le x^2 < 10\}$ on the $x$-axis, and Species B is confined to the region defined by set $Q = \{y \in \mathbb{Z} \mid |y-1| \le 1\}$ on the $y$-axis. The total search area is represented by the Cartesian product $P \times Q$. If a camera trap is placed at every point $(x, y)$ in $P \times Q$, how many camera traps are required in total?

A mapping department uses sets $M = \{x \in \mathbb{Z} : 1 \le x \le 5\}$ and $N = \{y \in \mathbb{Z} : 10 \le y \le 12\}$. They define a 'proximity' set $S = \{(x, y) \in M \times N : x + y > 15\}$. How many ordered pairs are in $S$?

If $A = \{1, 2\}$ and $B = \{3, 4\}$, then $A \times B$ is a set of points. If we consider the diagonal of the rectangle formed by these four points in the Cartesian plane, what is the length of this diagonal?

A relation $R$ is defined on the set of integers $Z$ such that $(x, y) \in R$ if and only if $|x - y| \leq 1$. If we observe a subset of this relation where $x \in \{1, 2, 3\}$, which of the following diagrams correctly represents the mapping of these elements to their possible images $y$ in the set $Y = \{1, 2, 3, 4\}$?

If the relation $R$ maps elements to their cubes, $y = x^3$, find the image of $x = -1$ using the cubic curve provided.

The provided graph represents $y = |x|$. For $x = -2$, what is the corresponding value of $y$ in the relation?

Look at the mapping from Set $M$ to Set $N$. If Set $M = \{10, 20\}$ and Set $N = \{5, 10\}$ such that $y = x/2$, what is the ordered pair for the first element?

Consider the function $f(x) = x^2 - 2$ defined for the domain $x \in \{-2, -1, 0, 1, 2\}$. The graph of this function consists of discrete points. Based on the provided coordinate geometry plot of $y = x^2 - 2$, find the range of this function.

Consider the step function shown. What is the value of the relation at $x = 1.5$?

Based on the mapping $f: x \to x^2 + 1$, if the codomain is as shown in the diagram, what is the value of the relation at $x = -2$?

Consider a set $S = \{x : x \text{ is a positive integer } < 10\}$. A relation $R$ is defined on $S$ by $R = \{(x, y) : y = 3x \text{ and } y \in S\}$. The mapping is visualized. What is the sum of all elements in the domain of $R$?

In a factory, the relation between the number of machines $m$ and the total output $O$ is $O = m^2 + m$. The set of machines available is $M = \{2, 3, 4, 5\}$. If a relation $R$ is defined as $R = \{(m, O) : m \in M\}$, and we look for pairs where $O$ is divisible by $10$, which machines satisfy this?

A set $A = \{1, 2, 3, 4, 5\}$ and a relation $R$ on $A$ is defined as $R = \{(x, y) : x^2 - y^2 = 5\}$. Based on the visualization of squares of these numbers, what is the cardinality of the relation $R$?

The graph of the function $h(x) = x^2 - 1$ is shown. At which of the following values of $x$ does the function cross the x-axis?

Consider the mapping between two sets $A$ and $B$ shown in the diagram. If this mapping represents a function $g(x)$, find the value of $g(3)$.

A linear function is defined by the equation $f(x) = 2x + 3$. If the domain of the function is restricted to $\{x : -2 \le x \le 2\}$, what is the range of the function?

If $f(x) = \sqrt{x}$, for what value of $x$ is $f(x) = 4$?

The function $p(x)$ is represented by the curve in the diagram. If $p(x) = k$ has exactly one solution for $x$ in the interval shown, what is the value of $k$ at the vertex?

Given the function $f(x) = x^2 + 2x + 1$, calculate the value of $f(a+1) - f(a)$.

For the function $f(x) = 1 - x^2$, what is the maximum value of the function?

A technician observes the temperature variation of a chemical substance in a reactor over a 6-hour period. The temperature $T(x)$ in degrees Celsius at time $x$ hours is modeled by the function $T(x) = |x - 3| + 2$. However, due to external cooling, the actual temperature recorded is a transformation $g(x) = T(x) - 1$. Based on the graph of the function $g(x)$ shown, identify the minimum temperature reached by the substance and the time at which it occurs.

A mapping $f$ from set $A = \{1, 2, 3\}$ to set $B = \{4, 5, 6\}$ is defined such that $f(1) = 4$, $f(2) = 5$, and $f(3) = 6$. A second mapping $g$ from $B$ to $C = \{7, 8, 9\}$ is defined by $g(y) = y + 3$. Find the output of the combined relation $g(f(2))$.

A light beam follows a path described by $g(x) = \frac{1}{x}$ for $x > 0$. As the horizontal distance $x$ becomes very large, what value does the vertical position $g(x)$ approach?

The graph of the identity function $f(x) = x$ is shown in the coordinate plane. Which of the following points must lie on this graph?

Based on the graph of $f(x) = |x| + 1$, what is the minimum value (y-coordinate) of the function?

In the diagram showing the Greates Integer Function $f(x) = [x]$, what is the value of $f(1.5)$?

Given the graph of $f(x) = x^3$, find the value of $f(-2)$.

The flow diagram represents the composition of two functions. If input $x=3$ goes through $A: x \to x+2$ and then $B: x \to x^2$, what is the final output $Z$?

What is the value of $k$ if the function $f(x) = x^2 + k$ passes through the point $(1, 5)$ as shown in the diagram?

Based on the transformation in the diagram, what is the equation of the red dashed curve if the blue solid curve is $f(x) = x^2$?

An engineer is analyzing a structural cable that follows the curve $y = 0.1x^2$. To reinforce it, a straight support beam $y = x - 2.5$ is added. At what $x$-coordinate(s) does the beam meet the cable? Inspect the graph for intersection points.

A robot moves along a floor following the path $y = |x - 2| + |x + 2|$. Based on the graph, what is the minimum $y$-coordinate reached by the robot and for what range of $x$ is this minimum maintained?

A designer creates a logo using the function $f(x) = \frac{1}{x}$. To ensure the logo fits within a specific frame, they only consider the interval $0.5 \le x \le 4$. What is the range of the function values (the heights) for this part of the logo?