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 ≥ 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 bakery scenario, x represents the number of batches, which must be whole numbers since you can't make a fraction of a batch in this context, and it ranges from 0 (no batches) to 20, making the realistic domain discrete integers. Choice B correctly identifies domain as x ∈ {0,1,2,…,20} based on the contextual constraints of whole batches and the daily limit. A common distractor like Choice A fails by assuming continuous reals, but batches aren't continuously variable—you can't make 3.5 batches here. 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 B(t) = $500(1.03)^t$, the exponential function mathematically accepts any real t (you can raise 1.03 to any power), but realistically t represents time since account opening, so t ≥ 0, and the account has been open at most 10 years, so t ≤ 10. Choice B correctly identifies mathematical domain as all real t (exponentials work for any input) and realistic domain as 0 ≤ t ≤ 10 (non-negative time up to 10 years). Choices A and C incorrectly restrict the mathematical domain, while choice D misunderstands the realistic constraint. 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.

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. The square root function $L(x) = \sqrt{x-2}$ mathematically requires $x \geq 2$ for real outputs, and contextually x is fencing from 2 to 50 meters, which is continuous as length can be any value in that range. Choice A correctly identifies domain as $x \in[2,50]$ based on both mathematical restriction and upper limit. Choice D fails by including x<2, where the square root is undefined or negative inside, which doesn't make sense. 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).

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 f(x) = √(x-4), the mathematical domain requires x - 4 ≥ 0, so x ≥ 4 (can't take square root of negative). Since x represents a physical length measurement, we'd normally add x ≥ 0, but x ≥ 4 already ensures this—the mathematical constraint is more restrictive than the physical one! Choice A correctly identifies both domains as x ≥ 4, recognizing that the square root requirement already enforces non-negativity and goes beyond it. Choices B, C, and D incorrectly analyze either the mathematical domain (square root needs x ≥ 4, not x > 4 or all reals) or misunderstand how constraints combine. 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.

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 fare model F(t) = 18/(6-t) + 3, mathematically t ≠ 6, but the context specifies only the first 5 hours, so realistic domain is [0,5], where the function is defined. Choice A correctly identifies domain as t ∈ [0,5] based on the intended use up to 5 hours. Choice D fails by extending to all reals except t=6, ignoring the contextual limit to 5 hours. 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).

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). 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 bike rental, time t can range from 0 minutes (no rental) to 180 minutes (maximum allowed), and since time is measured to the nearest second, it's continuous—you can rent for 45.5 minutes or 120.25 minutes. Choice B correctly identifies domain as [0,180] using interval notation for continuous values, including both endpoints since you can rent for exactly 0 or exactly 180 minutes. Choice A incorrectly treats time as discrete (only whole minutes), while choices C and D either ignore the upper limit or exclude valid endpoints. 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.

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(s) = s², the mathematical domain considers only what the formula allows: since we can square any real number (positive, negative, or zero), the mathematical domain is all real numbers. However, the realistic domain must consider that s represents a side length, which cannot be negative in the physical world, so s ≥ 0. Choice B correctly identifies mathematical domain as all real numbers (the formula s² accepts any input) and realistic domain as s ≥ 0 (side lengths must be non-negative). Choice A reverses these, C incorrectly excludes s = 0 from the mathematical domain, and D incorrectly restricts the 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.

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 C(w) = 5 + 2√(w-1), the mathematical domain requires w - 1 ≥ 0 (can't take square root of negative), so w ≥ 1. The context states packages must weigh at least 1 pound, which aligns perfectly with the mathematical constraint. Weight is continuous (packages can weigh 1.5 pounds, 3.7 pounds, etc.). Choice A correctly identifies domain as w ≥ 1 based on both the mathematical requirement (square root needs non-negative input) and the context constraint (minimum 1 pound weight). Choice B excludes w = 1 unnecessarily (√0 = 0 is valid), C ignores the square root constraint, and D incorrectly sets an upper bound. 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.