This question tests your understanding of how the domain of a function relates to both the context (what makes sense in real life) and the graph (how the function is represented visually). The domain is the set of all possible input values, and in real-world contexts, we need to think about what values actually make sense: if n represents the number of engines being assembled, we can't have n = -5 (negative engines) or n = 2.7 (partial engines), so the appropriate domain is positive integers {1, 2, 3, ...}, even though mathematically the formula might work for any number. In this scenario, n is the number of tickets sold in a theater with 200 seats, so n must be whole numbers from 0 (no tickets sold) to 200 (all seats filled), as you can't sell a fraction of a ticket or more than available seats. Choice C correctly identifies the domain as all integers n such that 0 ≤ n ≤ 200 because it accounts for the countable nature of tickets and the capacity limit. Choice A might seem appealing but allows non-integer values like n=1.5, which doesn't make sense for selling tickets. When determining domain from context, ask yourself: (1) What does the variable represent? (2) Can it be negative? (usually not for quantities, time, etc.) (3) Are there upper limits? (capacity, time limits) (4) Must it be whole numbers? (counting people/items) or can it be any number? (measuring distance/time). Answer these questions and the domain becomes clear!

This question tests your understanding of how the domain of a function relates to both the context (what makes sense in real life) and the graph (how the function is represented visually). Reading domain from a graph: look at which x-values have corresponding points on the graph. If the graph extends from x = 0 to x = 5 with a continuous curve, the domain is [0, 5]. If it shows separate dots at x = 0, 1, 2, 3, 4, 5, the domain is {0, 1, 2, 3, 4, 5}—the graph itself tells you the domain! Here, the solid line segment starts at a closed point at x = -1 (included) and ends at an open point at x = 4 (excluded), so the domain covers all x from -1 to just before 4. Choice A correctly identifies the domain as [-1,4) because it matches the inclusion of -1 and exclusion of 4 as shown. Choice C includes 4, but the open point means it's not part of the graph. Graph clue: if you see a solid curve or line, the domain is an interval (continuous). If you see separated dots, the domain is a list of specific values (discrete). The visual representation directly shows whether the function is defined everywhere in an interval or just at certain points!

This question tests your understanding of how the domain of a function relates to both the context (what makes sense in real life) and the graph (how the function is represented visually). The domain is the set of all possible input values, and in real-world contexts, we need to think about what values actually make sense: if n represents the number of engines being assembled, we can't have n = -5 (negative engines) or n = 2.7 (partial engines), so the appropriate domain is positive integers {1, 2, 3, ...}, even though mathematically the formula might work for any number. For the bakery packaging time P(n) = 60/n, n is the number of workers, which must be positive integers since you can't have zero, negative, or fractional workers, and the formula divides by n. Choice B correctly identifies the domain as {n | n is an integer and n ≥ 1} because it fits the real-life constraint of whole, positive numbers of people. Choice C allows fractions like 1.5 workers, which doesn't make practical sense here. When determining domain from context, ask yourself: (1) What does the variable represent? (2) Can it be negative? (usually not for quantities, time, etc.) (3) Are there upper limits? (capacity, time limits) (4) Must it be whole numbers? (counting people/items) or can it be any number? (measuring distance/time). Answer these questions and the domain becomes clear!

This question tests your understanding of how the domain of a function relates to both the context (what makes sense in real life) and the graph (how the function is represented visually). The domain is the set of all possible input values, and in real-world contexts, we need to think about what values actually make sense: if n represents the number of attendees in a 200-seat theater, we can't have n = -5 (negative attendees) or n = 150.5 (partial people), so the appropriate domain is integers from 0 to 200, even though mathematically the formula might work for any number. The theater context gives us two key constraints: attendees must be whole people (integers) and the theater has a maximum capacity of 200 seats. Choice B correctly identifies the domain as all integers from 0 to 200: {0, 1, 2, ..., 200} because you count people in whole numbers and the theater can have anywhere from 0 (empty) to 200 (full) attendees. Choice A [0, 200] would incorrectly allow fractional attendees like 125.7 people, which is impossible in real life. When determining domain from context, ask yourself: (1) What does the variable represent? (2) Can it be negative? (usually not for quantities, time, etc.) (3) Are there upper limits? (capacity, time limits) (4) Must it be whole numbers? (counting people/items) or can it be any number? (measuring distance/time). Answer these questions and the domain becomes clear!

This question tests your understanding of how the domain of a function relates to both the context (what makes sense in real life) and the graph (how the function is represented visually). The domain is the set of all possible input values, and in real-world contexts, we need to think about what values actually make sense: if n represents the number of text messages sent, we can't have n = -10 (negative messages) or n = 5.5 (partial messages), so the appropriate domain is non-negative integers {0, 1, 2, 3, ...}, even though mathematically the formula might work for any number. Text messages are discrete items that are counted in whole numbers, and it's possible to send 0 messages in a month (just paying the base fee). Choice C correctly identifies the domain as non-negative integers {0, 1, 2, 3, ...} because messages are counted in whole units starting from zero. Choice A (all real numbers n ≥ 0) would incorrectly allow fractional messages like 15.7 messages, which isn't possible when counting actual text messages sent. When determining domain from context, ask yourself: (1) What does the variable represent? (2) Can it be negative? (usually not for quantities, time, etc.) (3) Are there upper limits? (capacity, time limits) (4) Must it be whole numbers? (counting people/items) or can it be any number? (measuring distance/time). Answer these questions and the domain becomes clear!

This question tests your understanding of how the domain of a function relates to both the context (what makes sense in real life) and the graph (how the function is represented visually). The domain is the set of all possible input values, and in real-world contexts, we need to think about what values actually make sense: if n represents the number of engines being assembled, we can't have n = -5 (negative engines) or n = 2.7 (partial engines), so the appropriate domain is positive integers {1, 2, 3, ...}, even though mathematically the formula might work for any number. For the streaming service, m represents whole months since signing up, starting from 0 (initial sign-up with no months passed) and increasing as integers with no upper limit specified. Choice D correctly identifies the domain as all integers m such that m ≥ 0 because months are counted in whole units and can't be negative. Choice C allows real numbers, but you can't have a fraction of a month in this billing context. When determining domain from context, ask yourself: (1) What does the variable represent? (2) Can it be negative? (usually not for quantities, time, etc.) (3) Are there upper limits? (capacity, time limits) (4) Must it be whole numbers? (counting people/items) or can it be any number? (measuring distance/time). Answer these questions and the domain becomes clear!

This question tests your understanding of how the domain of a function relates to both the context (what makes sense in real life) and the graph (how the function is represented visually). When graphing functions, the domain determines whether we use a continuous line or curve (for measurable quantities like time, distance, or temperature) or discrete points (for countable quantities like number of people, items sold, or days). If you're counting things that come in whole units, use dots; if you're measuring continuous quantities, use a connected line or curve! In this ride-share scenario, miles can be measured continuously (you can travel 2.5 miles, 3.14 miles, etc.), and you can't travel negative miles, so m must be non-negative. Choice B correctly identifies the domain as all real numbers m ≥ 0 because distance is continuous and non-negative. Choice C is incorrect because it restricts miles to integers only (0, 1, 2, 3 miles), missing fractional distances like 1.7 miles that are perfectly valid for a car trip. For discrete vs continuous graphing: if the context involves counting distinct objects (tickets sold, students in class, days of the week), use separate dots because you can't have fractional amounts. If it involves measuring (time passing, distance traveled, temperature), use a connected line because the quantity can take any value in between. Think: 'Can there be in-between values?' If no, discrete. If yes, continuous!

This question tests your understanding of how the domain of a function relates to both the context (what makes sense in real life) and the graph (how the function is represented visually). The domain is the set of all possible input values, and in real-world contexts, we need to think about what values actually make sense: if $n$ represents the number of engines being assembled, we can't have $n = -5$ (negative engines) or $n = 2.7$ (partial engines), so the appropriate domain is positive integers ${1, 2, 3, \dots}$, even though mathematically the formula might work for any number. In this smoothie shop scenario, $n$ represents the number of smoothies made in a day, which must be whole numbers because you can't make a fraction of a smoothie, and it ranges from 0 (no smoothies) to 120 (maximum capacity), including 0 since the fixed cost applies even without making any. Choice B correctly identifies the domain as ${n \mid 0 \le n \le 120, n \text{ is an integer}}$ because it accounts for the contextual constraints of non-negative integers up to the shop's limit. Choice A is close but fails gently because it allows fractional values like 2.5 smoothies, which don't make sense in reality. When determining domain from context, ask yourself: (1) What does the variable represent? (2) Can it be negative? (usually not for quantities, time, etc.) (3) Are there upper limits? (capacity, time limits) (4) Must it be whole numbers? (counting people/items) or can it be any number? (measuring distance/time). Answer these questions and the domain becomes clear!

This question tests your understanding of how the domain of a function relates to both the context (what makes sense in real life) and the graph (how the function is represented visually). When graphing functions, the domain determines whether we use a continuous line or curve (for measurable quantities like time, distance, or temperature) or discrete points (for countable quantities like number of people, items sold, or days). If you're counting things that come in whole units, use dots; if you're measuring continuous quantities, use a connected line or curve! In this case, n represents the number of T-shirts, which must be whole numbers, so the graph should show separate points at integer values rather than a continuous line. Choice B correctly identifies that it should be separate points because n must be a whole number of shirts. Choice A suggests a continuous line, but that would imply fractional shirts, which isn't realistic for ordering. For discrete vs continuous graphing: if the context involves counting distinct objects (tickets sold, students in class, days of the week), use separate dots because you can't have fractional amounts. If it involves measuring (time passing, distance traveled, temperature), use a connected line because the quantity can take any value in between. Think: 'Can there be in-between values?' If no, discrete. If yes, continuous!

This question tests your understanding of how the domain of a function relates to both the context (what makes sense in real life) and the graph (how the function is represented visually). When graphing functions, the domain determines whether we use a continuous line or curve (for measurable quantities like time, distance, or temperature) or discrete points (for countable quantities like number of people, items sold, or days). If you're counting things that come in whole units, use dots; if you're measuring continuous quantities, use a connected line or curve! In this ball-throwing situation, t represents time in seconds, which is a continuous quantity from the launch at t=0 until it hits the ground at t=3, including both endpoints as the height is defined there. Choice A correctly identifies the domain as [0,3] because time flows continuously without gaps in this physical context. Choice B doesn't quite fit because it treats time as discrete integers, but time isn't limited to whole seconds here. For discrete vs continuous graphing: if the context involves counting distinct objects (tickets sold, students in class, days of the week), use separate dots because you can't have fractional amounts. If it involves measuring (time passing, distance traveled, temperature), use a connected line because the quantity can take any value in between. Think: 'Can there be in-between values?' If no, discrete. If yes, continuous!