Sets and Functions

Identify the set $B$ represented on the real number line below, where the endpoints are as shown.

A universal set $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ is represented by the rectangle in the Venn diagram. The circle represents set $A$. Find the cardinality of the complement set $A'$.

Let $A$ be the set of all triangles in a plane. What is the most appropriate universal set for $A$?

Given the set $B = \{x : x^2 - 1 = 0, x \in \mathbb{R}\}$, how many elements does it contain?

A set $S$ is defined as the domain of the function $f(x) = \sqrt{9 - x^2}$. Which of the following intervals represents the set $S$ as shown by the shaded region on the x-axis of the graph?

Two sets $P$ and $Q$ are represented in the Venn diagram below. If $n(P) = 15$, $n(Q) = 12$, and $n(P \cup Q) = 22$, determine the number of elements in the shaded region $P \cap Q$.

If $A \subseteq B$, which of the following diagrams correctly represents this relationship?

A researcher is mapping out the inhabitants of three remote islands: $A$, $B$, and $C$. The relationships between the sets of inhabitants are illustrated in the Venn diagram. Island $A$ contains 60 distinct species, Island $B$ contains 55, and Island $C$ contains 50. The number of species common to $A$ and $B$ is 15, to $B$ and $C$ is 12, and to $A$ and $C$ is 10. If the total number of unique species across all three islands is 135, how many species are found on all three islands simultaneously?

The population of two bacterial cultures $A$ and $B$ are modeled by sets of points in a plane. Culture $A$ is inside the circle $x^2 + y^2 = 9$ and culture $B$ is inside the circle $(x-3)^2 + y^2 = 9$. What is the set of points representing their overlap $A \cap B$?

A logistics firm uses a mapping $f: A \to B$ where $A = \{1, 2, 3\}$ and $B = \{4, 5\}$. How many onto (surjective) functions can be defined from set $A$ to set $B$?

A set with no elements is called a/an:

If $A = \{x, y, z\}$, how many subsets does set $A$ have?

Find the value of $n(P - Q)$ if $n(P) = 15$ and $n(P \cap Q) = 5$.

What is the domain of the function shown in the graph $f(x) = \sqrt{x}$?

In a survey of 100 students regarding their preference for subjects, $50$ students like Mathematics ($M$), $40$ students like Physics ($P$), and $10$ students like neither. Using the Venn diagram representation where the rectangle represents the universal set $U$ and the circles represent sets $M$ and $P$, find the number of students who like both Mathematics and Physics, denoted by $n(M \cap P)$.

If $f(x) = x^2$ and $g(x) = x + 2$, find the intersection point(s) of the two functions as shown in the graph.

In a group of 80 people, 45 speak Hindi, 30 speak English, and 15 speak both. How many people speak only Hindi?

Given two sets $A = \{x : x^2 - 5x + 6 = 0\}$ and $B = \{x : x^2 - 3x + 2 = 0\}$. If $U = \{1, 2, 3, 4, 5\}$, find the complement of $(A \cup B)$.

A library has 300 members. 140 read Fiction, 120 read Non-fiction, and 150 read Magazines. 40 read Fiction and Non-fiction, 50 read Non-fiction and Magazines, and 60 read Fiction and Magazines. 30 read all three. How many members read exactly two of the three types?

In a research center, 45 scholars study Physics, 50 study Mathematics, and 35 study Chemistry. 15 study Physics and Mathematics, 12 study Mathematics and Chemistry, and 10 study Physics and Chemistry. If 5 scholars study all three, find the number of scholars who study at most one of these subjects, assuming there are 100 scholars in the center.

In the given diagram, which set represents the range of the relation?

Identify the range of the function $f(x) = -x$ from the graph.

For the function $f(x) = x + 1$ shown, if the domain is restricted to $\{1, 2, 3\}$, what is the range?

Look at the mapping. Is this relation a function?

Find the domain of $f(x) = \frac{1}{\sqrt{x}}$ from its graph.

Given the floor function graph $f(x) = \lfloor x \rfloor$ (greatest integer function), what is the range?

If a relation $R = \{(x, y) : x^2 + y^2 = 9\}$ is graphed, what is the domain?

A telecommunications company models the signal strength $S$ of a satellite beam as a function of the horizontal distance $x$ (in km) from the base station. The relation is defined by the function $f(x) = \sqrt{16 - x^2} + \frac{1}{x-2}$. If the domain of this relation is the set of all real values of $x$ for which the signal is physically measurable and finite, determine the domain of $f$.

An urban planner maps the elevation of a hill using $E(x) = 5 - |x|$. The planner needs to know the domain for which the elevation is non-negative ($E(x) \ge 0$). Based on the triangle-like profile in the diagram, what is the domain?

A chemist observes a reaction where the pressure $P$ varies with volume $V$ as $P = \frac{10}{V}$. If the volume $V$ is constrained between 2 and 10 units as shown in the graph, what is the range of the pressure $P$?

Given the function $f(x) = |x - 2| + 1$, identify the range of the function based on its graph provided below.

What is the value of $f(-2)$ for the function $f(x) = x^2$ shown?

Identify the value of $x$ for which $f(x) = 0$ in the given graph of $f(x) = x - 4$.

The graph of $f(x) = x^3$ is shown. Is this function one-to-one (injective)?

Determine the range of the function $f(x) = \sin^2(x)$ as shown in the plot.

The step function shown is the Greatest Integer Function $f(x) = [x]$. What is the value of $f(1.5)$?

Based on the transformation shown, if the original graph is $f(x) = x^2$, what is the equation of the shifted graph $g(x)$?

A particle moves in a 1D space with potential energy $U(x) = \frac{1}{x^2 - x - 6}$. Identifying the 'forbidden' regions where the energy goes to infinity, find the domain of the potential energy function.

The voltage in a circuit is given by $V(t) = \log_{2}(t^2 - 4t + 5)$. What is the range of the voltage for all time $t \geq 0$?

A data encryption algorithm uses the function $f(x) = \sqrt{x^2 - 1} + \frac{1}{\sqrt{4 - x^2}}$. For which set of real values $x$ is the key generation valid?

Given two real functions $f(x) = x^2$ and $g(x) = x + 2$, find the value of the product function $(f \cdot g)(3)$.

Let $f(x) = x^2$ and $g(x) = x$. Calculate the value of the quotient function $\left(\frac{f}{g}\right)(x)$ at $x = 3$. Note that the domain of the quotient excludes $x=0$.

Identify the value of $(f - g)(0)$ if $f(x) = x^3 + 5$ and $g(x) = x^2 + 2$.

If $f(x) = 2x$ and $g(x) = 3$, find the function $(f/g)(x)$.

Let $f(x) = x^2 - 9$ and $g(x) = x - 3$. Calculate the value of $(f/g)(3)$.

If $f(x) = 2^x$ and $g(x) = 2^{-x}$, find the value of $(f + g)(1)$.

Determine $(f - g)(x)$ where $f(x) = \frac{1}{x}$ and $g(x) = x$.

Given two functions $f(x) = \frac{1}{x+2}$ and $g(x) = \frac{x}{x-3}$. Let $h(x) = (f \cdot g)(x)$. Determine the value of $x$ for which $h(x) = 1$.

The supply function for a commodity is $S(p) = p^2 + 2p + 10$ and the demand function is $D(p) = 200 - 3p - p^2$, where $p$ is the price. The market excess supply is defined as $E(p) = (S - D)(p)$. Find the price $p$ at which the market is in equilibrium (where excess supply is zero).

Two harmonic waves are traveling in a medium. Their displacements are given by $y_1(x) = 3 \sin(x)$ and $y_2(x) = 4 \cos(x)$. The resulting wave is the sum $Y(x) = (y_1 + y_2)(x)$. Find the maximum amplitude of the resultant wave.