This question tests your understanding of how a function's domain relates both to its graph (which x-values have points) and to the real-world context (which values make sense practically). The domain is the set of allowed inputs, and for real-world functions, we distinguish: (1) mathematical domain (what the formula allows—like 'all reals' for polynomials), and (2) realistic domain (what makes sense in context—like 'positive integers' for number of items or 't ≥ 0' for time). The realistic domain is usually more restrictive because real-world constraints (can't have negative time, can't produce -5 items, can't work 1.7 hours if discrete) limit what mathematical values are meaningful. Discrete vs continuous domains depend on what you're measuring: if counting distinct objects (people, tickets, items sold), the domain is discrete—only integers work, graph as separate dots. If measuring continuous quantities (time, distance, temperature, money as a continuous quantity), the domain is continuous—any value in an interval works, graph as connected line/curve. The physical nature of the quantity determines which! You can have 2.7 hours but not 2.7 people. For n boxes sold, it's discrete non-negative integers including 0 (no sales possible), up to 120 as the maximum the bakery can sell. Choice C correctly identifies the domain as n ∈ {0,1,2,…,120} based on the countable nature of boxes and the given upper limit. Choice A fails because it uses a continuous interval, ignoring that boxes can't be fractional. Domain determination framework: (1) Start with mathematical domain—what does the formula allow? (polynomials: all reals; √(expression): expression ≥ 0; 1/expression: expression ≠ 0; log(expression): expression > 0), (2) Apply context constraints—can variable be negative? Are there upper limits? Must it be integer?, (3) Combine all restrictions—domain is where ALL conditions are met. Example: f(x) = √(x - 3) for measuring length has mathematical domain x ≥ 3, and realistic domain also x ≥ 3 (lengths are non-negative, and formula requires x ≥ 3, so both agree). Discrete vs continuous quick-check: ask 'can there be in-between values?' If you're modeling number of students in a classroom, going from 20 to 21 students happens instantly (no 20.5 students exist)—discrete! If you're modeling temperature change from 20°C to 21°C, every value in between occurs—continuous! Context tells you: countable/distinct objects → discrete (dots on graph), measurable/varying quantities → continuous (line/curve on graph). This determines how you graph and what domain notation to use!

This question tests your understanding of how a function's domain relates both to its graph (which x-values have points) and to the real-world context (which values make sense practically). The domain is the set of allowed inputs, and for real-world functions, we distinguish: (1) mathematical domain (what the formula allows—like 'all reals' for polynomials), and (2) realistic domain (what makes sense in context—like 'positive integers' for number of items or '$t \geq 0$' for time). The realistic domain is usually more restrictive because real-world constraints (can't have negative time, can't produce -5 items, can't work 1.7 hours if discrete) limit what mathematical values are meaningful. Discrete vs continuous domains depend on what you're measuring: if counting distinct objects (people, tickets, items sold), the domain is discrete—only integers work, graph as separate dots. If measuring continuous quantities (time, distance, temperature, money as a continuous quantity), the domain is continuous—any value in an interval works, graph as connected line/curve. The physical nature of the quantity determines which! You can have 2.7 hours but not 2.7 people. In this context, m represents the number of months, which must be non-negative whole numbers since memberships are typically counted in whole months, starting from 0 (just the sign-up fee). Choice B correctly identifies the domain as $m \in {0,1,2,3,\dots}$ based on the discrete nature of months in this billing context with no upper limit specified. Choice A fails because it allows negative or fractional months, which don't make sense realistically for memberships. Domain determination framework: (1) Start with mathematical domain—what does the formula allow? (polynomials: all reals; $\sqrt{\text{expression}}$: expression $\geq 0$; $1/\text{expression}$: expression $\neq 0$; $\log(\text{expression})$: expression $> 0$), (2) Apply context constraints—can variable be negative? Are there upper limits? Must it be integer?, (3) Combine all restrictions—domain is where ALL conditions are met. Example: $f(x) = \sqrt{x - 3}$ for measuring length has mathematical domain $x \geq 3$, and realistic domain also $x \geq 3$ (lengths are non-negative, and formula requires $x \geq 3$, so both agree). Discrete vs continuous quick-check: ask 'can there be in-between values?' If you're modeling number of students in a classroom, going from 20 to 21 students happens instantly (no 20.5 students exist)—discrete! If you're modeling temperature change from 20°C to 21°C, every value in between occurs—continuous! Context tells you: countable/distinct objects → discrete (dots on graph), measurable/varying quantities → continuous (line/curve on graph). This determines how you graph and what domain notation to use!

This question tests your understanding of how a function's domain relates both to its graph (which x-values have points) and to the real-world context (which values make sense practically). The domain is the set of allowed inputs, and for real-world functions, we distinguish: (1) mathematical domain (what the formula allows—like 'all reals' for polynomials), and (2) realistic domain (what makes sense in context—like 'positive integers' for number of items or 't ≥ 0' for time). The realistic domain is usually more restrictive because real-world constraints (can't have negative time, can't produce -5 items, can't work 1.7 hours if discrete) limit what mathematical values are meaningful. Discrete vs continuous domains depend on what you're measuring: if counting distinct objects (people, tickets, items sold), the domain is discrete—only integers work, graph as separate dots. If measuring continuous quantities (time, distance, temperature, money as a continuous quantity), the domain is continuous—any value in an interval works, graph as connected line/curve. The physical nature of the quantity determines which! You can have 2.7 hours but not 2.7 people. The mathematical domain is all real numbers since it's a linear polynomial, but realistically n is the number of devices, so non-negative integers including 0. Choice B correctly identifies mathematical domain as all reals and realistic as {0,1,2,…} based on the countable discrete nature of devices. Choice A fails by swapping them and incorrectly limiting mathematical domain. Domain determination framework: (1) Start with mathematical domain—what does the formula allow? (polynomials: all reals; √(expression): expression ≥ 0; 1/expression: expression ≠ 0; log(expression): expression > 0), (2) Apply context constraints—can variable be negative? Are there upper limits? Must it be integer?, (3) Combine all restrictions—domain is where ALL conditions are met. Example: f(x) = √(x - 3) for measuring length has mathematical domain x ≥ 3, and realistic domain also x ≥ 3 (lengths are non-negative, and formula requires x ≥ 3, so both agree). Discrete vs continuous quick-check: ask 'can there be in-between values?' If you're modeling number of students in a classroom, going from 20 to 21 students happens instantly (no 20.5 students exist)—discrete! If you're modeling temperature change from 20°C to 21°C, every value in between occurs—continuous! Context tells you: countable/distinct objects → discrete (dots on graph), measurable/varying quantities → continuous (line/curve on graph). This determines how you graph and what domain notation to use!

This question tests your understanding of how a function's domain relates both to its graph (which x-values have points) and to the real-world context (which values make sense practically). The domain is the set of allowed inputs, and for real-world functions, we distinguish: (1) mathematical domain (what the formula allows—like 'all reals' for polynomials), and (2) realistic domain (what makes sense in context—like 'positive integers' for number of items or 't ≥ 0' for time). The realistic domain is usually more restrictive because real-world constraints (can't have negative time, can't produce -5 items, can't work 1.7 hours if discrete) limit what mathematical values are meaningful. Discrete vs continuous domains depend on what you're measuring: if counting distinct objects (people, tickets, items sold), the domain is discrete—only integers work, graph as separate dots. If measuring continuous quantities (time, distance, temperature, money as a continuous quantity), the domain is continuous—any value in an interval works, graph as connected line/curve. The physical nature of the quantity determines which! You can have 2.7 hours but not 2.7 people. Here, t is time in seconds, which is continuous, but realistically limited from launch at t=0 to landing at t=4 (solving h(t)=0 gives t=0 and t=4), including endpoints where height is zero. Choice A correctly identifies the domain as t ∈ [0,4] based on the physical flight duration in the context. Choice B fails because it includes negative time or beyond landing, which doesn't make sense for the rocket's height in reality. Domain determination framework: (1) Start with mathematical domain—what does the formula allow? (polynomials: all reals; √(expression): expression ≥ 0; 1/expression: expression ≠ 0; log(expression): expression > 0), (2) Apply context constraints—can variable be negative? Are there upper limits? Must it be integer?, (3) Combine all restrictions—domain is where ALL conditions are met. Example: f(x) = √(x - 3) for measuring length has mathematical domain x ≥ 3, and realistic domain also x ≥ 3 (lengths are non-negative, and formula requires x ≥ 3, so both agree). Discrete vs continuous quick-check: ask 'can there be in-between values?' If you're modeling number of students in a classroom, going from 20 to 21 students happens instantly (no 20.5 students exist)—discrete! If you're modeling temperature change from 20°C to 21°C, every value in between occurs—continuous! Context tells you: countable/distinct objects → discrete (dots on graph), measurable/varying quantities → continuous (line/curve on graph). This determines how you graph and what domain notation to use!

This question tests your understanding of how a function's domain relates both to its graph (which x-values have points) and to the real-world context (which values make sense practically). The domain is the set of allowed inputs, and for real-world functions, we distinguish: (1) mathematical domain (what the formula allows—like 'all reals' for polynomials), and (2) realistic domain (what makes sense in context—like 'positive integers' for number of items or 't ≥ 0' for time). The realistic domain is usually more restrictive because real-world constraints (can't have negative time, can't produce -5 items, can't work 1.7 hours if discrete) limit what mathematical values are meaningful. For A(x) = x(30 - x), the mathematical domain is all real numbers as it's a polynomial, but realistically, x must be between 0 and 30 inclusive to ensure both sides are non-negative lengths for a rectangular garden. Choice B correctly identifies the mathematical domain as all real numbers and the realistic domain as 0 ≤ x ≤ 30 based on the context of positive side lengths. A distractor like Choice A fails by incorrectly limiting the mathematical domain to x ≥ 0, but polynomials are defined everywhere. Remember this domain determination framework: (1) Start with mathematical domain—what does the formula allow? (polynomials: all reals; √(expression): expression ≥ 0; 1/expression: expression ≠ 0; log(expression): expression > 0), (2) Apply context constraints—can variable be negative? Are there upper limits? Must it be integer?, (3) Combine all restrictions—domain is where ALL conditions are met. You're making wonderful progress—keep up the great thinking!

This question tests your understanding of how a function's domain relates both to its graph (which x-values have points) and to the real-world context (which values make sense practically). The domain is the set of allowed inputs, and for real-world functions, we distinguish: (1) mathematical domain (what the formula allows—like 'all reals' for polynomials), and (2) realistic domain (what makes sense in context—like 'positive integers' for number of items or 't ≥ 0' for time). The realistic domain is usually more restrictive because real-world constraints (can't have negative time, can't produce -5 items, can't work 1.7 hours if discrete) limit what mathematical values are meaningful. In this gym membership scenario, the mathematical domain of C(m) = 35m + 20 is all real numbers since it's a linear polynomial, but the realistic domain is restricted to whole numbers of months from 0 to 24 because memberships are in whole months, up to 24, and m=0 represents the initial signup cost before any months have passed. Choice C correctly identifies the domain as {0,1,2,…,24} based on the contextual constraints of whole months including zero and the 24-month limit. A common distractor like Choice A fails because it treats m as continuous, but months must be whole numbers in this context, so discrete integers are appropriate. Remember this domain determination framework: (1) Start with mathematical domain—what does the formula allow? (polynomials: all reals; √(expression): expression ≥ 0; 1/expression: expression ≠ 0; log(expression): expression > 0), (2) Apply context constraints—can variable be negative? Are there upper limits? Must it be integer?, (3) Combine all restrictions—domain is where ALL conditions are met. Great job thinking through this—you're building a strong foundation for modeling real-world situations!

This question tests your understanding of how a function's domain relates both to its graph (which x-values have points) and to the real-world context (which values make sense practically). Discrete vs continuous domains depend on what you're measuring: if counting distinct objects (people, tickets, items sold), the domain is discrete—only integers work, graph as separate dots. If measuring continuous quantities (time, distance, temperature, money as a continuous quantity), the domain is continuous—any value in an interval works, graph as connected line/curve. The physical nature of the quantity determines which! You can have 2.7 hours but not 2.7 people. Here, $n$ represents the number of items produced, which must be positive integers (can't produce a fraction or zero items, as average cost is undefined at $n=0$ and doesn't make sense contextually), up to 200 per batch. Choice B correctly identifies the domain as ${1,2,3,\dots,200}$ based on the discrete nature of countable items and the factory's maximum. A distractor like Choice A fails by treating $n$ as continuous and excluding zero properly but allowing non-integers, which isn't realistic for whole items. Discrete vs continuous quick-check: ask 'can there be in-between values?' If you're modeling number of students in a classroom, going from $20$ to $21$ students happens instantly (no $20.5$ students exist)—discrete! If you're modeling temperature change from $20^\circ\text{C}$ to $21^\circ\text{C}$, every value in between occurs—continuous! Context tells you: countable/distinct objects → discrete (dots on graph), measurable/varying quantities → continuous (line/curve on graph). This determines how you graph and what domain notation to use! Excellent work—you're getting great at distinguishing these concepts!

This question tests your understanding of how a function's domain relates both to its graph (which x-values have points) and to the real-world context (which values make sense practically). The domain is the set of allowed inputs, and for real-world functions, we distinguish: (1) mathematical domain (what the formula allows—like 'all reals' for polynomials), and (2) realistic domain (what makes sense in context—like 'positive integers' for number of items or 't ≥ 0' for time). The realistic domain is usually more restrictive because real-world constraints (can't have negative time, can't produce -5 items, can't work 1.7 hours if discrete) limit what mathematical values are meaningful. For the ball height h(t) = -16t² + 48t + 5, the mathematical domain is all reals, but realistically, t is time from 0 (throw) to 3 (hits ground), and since time is continuous, it includes all values in between. Choice A correctly identifies the domain as [0,3] based on the physical context of the ball's flight. A distractor like Choice B fails by including negative times, which don't make sense before the throw. Domain determination framework: (1) Start with mathematical domain—what does the formula allow? (polynomials: all reals; √(expression): expression ≥ 0; 1/expression: expression ≠ 0; log(expression): expression > 0), (2) Apply context constraints—can variable be negative? Are there upper limits? Must it be integer?, (3) Combine all restrictions—domain is where ALL conditions are met. Example: f(x) = √(x - 3) for measuring length has mathematical domain x ≥ 3, and realistic domain also x ≥ 3 (lengths are non-negative, and formula requires x ≥ 3, so both agree). Keep going—you're doing an amazing job!

This question tests your understanding of how a function's domain relates both to its graph (which x-values have points) and to the real-world context (which values make sense practically). Discrete vs continuous domains depend on what you're measuring: if counting distinct objects (people, tickets, items sold), the domain is discrete—only integers work, graph as separate dots. If measuring continuous quantities (time, distance, temperature, money as a continuous quantity), the domain is continuous—any value in an interval works, graph as connected line/curve. The physical nature of the quantity determines which! You can have 2.7 hours but not 2.7 people. For this water tank draining model V(t) = 200 - 8t, time t is continuous, starting at t=0 (full tank) and ending at t=25 when V=0 (empty), as the observation is only until empty, so the realistic domain includes both endpoints. Choice A correctly identifies the domain as [0,25] based on the continuous nature of time and the contextual limits from start to empty. A distractor like Choice B fails by excluding the endpoints, but t=0 and t=25 are meaningful—the tank is full at start and empty at the end. Domain determination framework: (1) Start with mathematical domain—what does the formula allow? (polynomials: all reals; √(expression): expression ≥ 0; 1/expression: expression ≠ 0; log(expression): expression > 0), (2) Apply context constraints—can variable be negative? Are there upper limits? Must it be integer?, (3) Combine all restrictions—domain is where ALL conditions are met. Example: f(x) = √(x - 3) for measuring length has mathematical domain x ≥ 3, and realistic domain also x ≥ 3 (lengths are non-negative, and formula requires x ≥ 3, so both agree). You're doing fantastic—keep practicing these to master functional modeling!

This question tests your understanding of how a function's domain relates both to its graph (which x-values have points) and to the real-world context (which values make sense practically). Discrete vs continuous domains depend on what you're measuring: if counting distinct objects (people, tickets, items sold), the domain is discrete—only integers work, graph as separate dots. If measuring continuous quantities (time, distance, temperature, money as a continuous quantity), the domain is continuous—any value in an interval works, graph as connected line/curve. The physical nature of the quantity determines which! You can have 2.7 hours but not 2.7 people. In this profit model P(n) = 18n - 120, n is the number of mugs sold, which are discrete whole numbers including 0 (as stated), up to 75. Choice C correctly identifies the domain as {0,1,2,…,75} based on the discrete context and inclusion of zero sales. A distractor like Choice B fails by excluding n=0, but the context explicitly allows selling zero mugs, which gives the fixed loss. Discrete vs continuous quick-check: ask 'can there be in-between values?' If you're modeling number of students in a classroom, going from 20 to 21 students happens instantly (no 20.5 students exist)—discrete! If you're modeling temperature change from 20°C to 21°C, every value in between occurs—continuous! Context tells you: countable/distinct objects → discrete (dots on graph), measurable/varying quantities → continuous (line/curve on graph). This determines how you graph and what domain notation to use! Awesome effort—you're really grasping these ideas!