Algebra Quiz: Write Explicit Or Recursive Functions

What this quiz covers

This quiz focuses on Write Explicit Or Recursive Functions, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. $h(t)=16t^2+24t+4$
2. $h(t)=-16t^2+24t+4$ (correct answer)
3. $h(t)=-16t^2+4t+24$
4. $h(t)=-24t^2+16t+4$

Explanation: This question tests your ability to translate a real-world situation into mathematical language—either as an explicit formula, a recursive process, or a series of calculation steps. The key to writing functions from context is identifying what changes (independent variable, like time or number of items) and what you're calculating (dependent variable, like cost or height), then finding the mathematical relationship between them using clues in the language. This context involves motion with gravity ('ball is thrown upward'), which tells us this is a quadratic function. The standard form for height with gravity is h(t) = -16t² + v₀t + h₀, where v₀ is initial velocity and h₀ is initial height. From the context we extract: initial height = 4 feet ('from a platform 4 feet high'), initial velocity = 24 ft/s ('initial upward velocity of 24 ft/s'), giving us h(t) = -16t² + 24t + 4. Choice A is correct because it properly includes the gravity term (-16t²), the initial velocity term (24t), and the initial height (4), matching the standard physics formula for projectile motion. Choice B has positive 16t² instead of negative, which would mean the ball accelerates upward forever—that violates the laws of physics! Gravity always pulls down, so the t² term must be negative in height functions. Context translation cheat sheet: 'per/each' = multiply by that rate, 'plus/in addition' = add, 'starts at' = initial value, 'doubles/triples' = multiply by 2 or 3 (exponential), 'percent' = divide by 100 for decimal. These phrases are your clues for turning words into math!

Question 2

1. $C(t)=0.05(t+25)$
2. $C(t)=25+0.05t$ (correct answer)
3. $C(t)=0.05+25t$
4. $C(t)=25t+0.05$

Explanation: This question tests your ability to translate a real-world situation into mathematical language—either as an explicit formula, a recursive process, or a series of calculation steps. An explicit expression gives you a direct formula to calculate the output from the input without needing any previous values: if the context says 'costs $45 per day plus$25 fee,' you can write C(d) = 45d + 25, where d is days and C is cost. Just plug in any number of days and calculate immediately! Looking at the context 'costs $25 each month plus$0.05 per text message,' we identify the independent variable as t (number of texts) and dependent variable as C (monthly cost). The relationship is linear because we have a fixed monthly cost plus a constant rate per text. The fixed monthly cost is $25 from 'costs$25 each month,' and the per-text rate is $0.05 from '$0.05 per text message.' Putting this together: C(t) = 25 + 0.05t. This formula lets us calculate monthly cost for any number of texts! Choice B is correct because it properly identifies t as the variable for texts and includes both the fixed monthly cost ($25) and the per-text rate ($0.05) in the standard form: fixed cost + rate × variable. Choice C swaps the coefficients: 0.05 + 25t would mean 5 cents fixed plus $25 per text—that would be an incredibly expensive texting plan! Always match the numbers to their meanings: 'per' indicates the rate that multiplies the variable. To write explicit functions from context: (1) Underline key phrases like 'per,' 'starts at,' 'plus,' 'each'—these tell you operations and values, (2) Identify what changes (independent variable x, n, t) and what you're calculating (dependent variable C, h, P), (3) Determine function type: constant rate = linear, percent/doubling = exponential, area/motion = quadratic, (4) Extract numbers from context and plug into the right form (y = mx + b for linear, y = a·b^x for exponential, etc.).

Question 3

1. $a_1=25,\; a_{n+1}=a_n+4$ (correct answer)
2. $a_1=25,\; a_{n+1}=4a_n$
3. $a_n=25+4n$
4. $a_1=4,\; a_{n+1}=a_n+25$

Explanation: This question tests your ability to translate a real-world situation into mathematical language—either as an explicit formula, a recursive process, or a series of calculation steps. A recursive definition describes how to get each term from the previous one, which is perfect for sequential processes: if you 'start with 200 and add 50 each week,' that becomes a₁ = 200, aₙ₊₁ = aₙ + 50. You need both the starting value and the rule for what comes next. The context describes a sequential process: 'starts a savings challenge with $25 on day 1 and adds$4 each new day.' We need a starting point—that's $25 from 'starts...with$25 on day 1'—and a rule for each step. Since we 'add $4 each new day,' each new value is the previous value plus 4. Writing this as a recursive definition: a₁ = 25, aₙ₊₁ = aₙ + 4. To find the 5th day's amount, we'd start at 25 and apply the rule four times! Choice A is correct because it includes both the initial value (a₁ = 25) and the recurrence relation (aₙ₊₁ = aₙ + 4) that matches the context of adding$4 each day. Choice B has the recursive rule backwards: it starts with $4 and adds$25 each time, but the context says we start with $25 and add$4. When writing recursive definitions, the starting value comes from 'starts with' or 'begins at,' not from the amount that changes! For recursive definitions, remember the two-part recipe: you MUST have both (1) the starting value(s)—look for 'starts at,' 'begins with,' 'initially'—and (2) the rule relating each term to the previous—look for 'add,' 'multiply by,' 'increases by' followed by a description. Write it as: a₁ = [starting value], aₙ₊₁ = [rule using aₙ]. If you're stuck choosing between explicit and recursive, ask: does each value only depend on which step you're at (explicit), or does each value depend on the previous value (recursive)? 'Day 5 costs $45' suggests explicit. 'Add$10 to yesterday's amount' suggests recursive. The language tells you which form fits!

Question 4

1. $a_1=40,\ a_{n+1}=a_n+15$ (correct answer)
2. $a_1=15,\ a_{n+1}=a_n+40$
3. $a_n=40+15n$
4. $a_{n+1}=15n+40$

Explanation: This question tests your ability to translate a real-world situation into mathematical language—either as an explicit formula, a recursive process, or a series of calculation steps. A recursive definition describes how to get each term from the previous one, which is perfect for sequential processes: if you 'start with 200 and add 50 each week,' that becomes a₁ = 200, aₙ₊₁ = aₙ + 50. You need both the starting value and the rule for what comes next. The context describes a sequential process: 'starts with $40' and 'adds$15 more than the previous week's total.' We need a starting point—that's $40 from 'starts with$40'—and a rule for each step. Since we 'add $15' each time, each new value is the previous value plus 15. Writing this as a recursive definition: a₁ = 40, aₙ₊₁ = aₙ + 15. To find the 5th week's amount, we'd start at 40 and add 15 four times! Choice A is correct because it includes both the initial value (a₁ = 40) and the recurrence relation (aₙ₊₁ = aₙ + 15) that matches adding$15 each week to the previous total. Choice B reverses the values: it starts with $15 and adds$40 each time, which doesn't match the problem description at all. When you see 'starts with X and adds Y each time,' X is your initial value and Y is what you add in the recursive rule! For recursive definitions, remember the two-part recipe: you MUST have both (1) the starting value(s)—look for 'starts at,' 'begins with,' 'initially'—and (2) the rule relating each term to the previous—look for 'add,' 'multiply by,' 'increases by' followed by a description. Write it as: a₁ = [starting value], aₙ₊₁ = [rule using aₙ].

Question 5

1. $S(n)=80(25)^n$
2. $S(n)=25+80n$
3. $S(n)=80n+25$
4. $S(n)=80+25n$ (correct answer)

Explanation: This question tests your ability to translate a real-world situation into mathematical language—either as an explicit formula, a recursive process, or a series of calculation steps. An explicit expression gives you a direct formula to calculate the output from the input without needing any previous values: if the context says 'costs $45 per day plus$25 fee,' you can write C(d) = 45d + 25, where d is days and C is cost. Just plug in any number of days and calculate immediately! Looking at the context 'score starts at 80 points and increases by 25 points each level completed,' we identify the independent variable as n (number of levels completed) and dependent variable as S(n) (score). The relationship is linear because we have a starting score plus a constant increase per level. The initial score is 80 from 'starts at 80 points,' and the rate is 25 from 'increases by 25 points each level.' Putting this together: S(n) = 80 + 25n. This formula lets us calculate the score after any number of levels! Choice B is correct because it properly identifies 80 as the starting score (when n = 0, no levels completed) and 25n as the points gained from completing n levels, matching the context perfectly. Choice C has the same mathematical result but writes it as 25 + 80n, which reverses the roles: this would mean starting at 25 points and gaining 80 per level, but the context clearly states we start at 80 and gain 25 per level. The order matters for understanding! Context translation cheat sheet: 'per/each' = multiply by that rate, 'plus/in addition' = add, 'starts at' = initial value, 'doubles/triples' = multiply by 2 or 3 (exponential), 'percent' = divide by 100 for decimal. These phrases are your clues for turning words into math!

Question 6

1. $h(t)=-16t^2-40t+6$
2. $h(t)=16t^2+40t+6$
3. $h(t)=-16t^2+40t+6$ (correct answer)
4. $h(t)=-40t^2+16t+6$

Explanation: This question tests your ability to translate a real-world situation into mathematical language—either as an explicit formula, a recursive process, or a series of calculation steps. The key to writing functions from context is identifying what changes (independent variable, like time or number of items) and what you're calculating (dependent variable, like cost or height), then finding the mathematical relationship between them using clues in the language. This context involves motion with gravity, which tells us this is a quadratic function. Quadratic functions model situations where acceleration is constant, like objects under gravity. The form is h(t) = -16t² + v₀t + h₀, and from the context we extract: gravity coefficient = -16 (standard for feet), initial velocity v₀ = 40 from 'initial upward velocity of 40 ft/s,' and initial height h₀ = 6 from 'platform 6 feet high' = h(t) = -16t² + 40t + 6. Choice B is correct because it includes the negative coefficient -16t² (gravity pulls down), the positive initial velocity term +40t (thrown upward), and the initial height +6, matching the standard form for projectile motion. Choice A has +16t² instead of -16t²: gravity always pulls objects down, which requires a negative coefficient on t². The negative sign is crucial for modeling falling objects! Context translation cheat sheet: 'per/each' = multiply by that rate, 'plus/in addition' = add, 'starts at' = initial value, 'doubles/triples' = multiply by 2 or 3 (exponential), 'percent' = divide by 100 for decimal. These phrases are your clues for turning words into math!

Question 7

1. $a_1=60,\; a_{n+1}=15a_n$
2. $a_1=15,\; a_{n+1}=a_n+60$
3. $a_1=60,\; a_{n+1}=a_n+15$ (correct answer)
4. $a_n=60+15n$

Explanation: This question tests your ability to translate a real-world situation into mathematical language—either as an explicit formula, a recursive process, or a series of calculation steps. A recursive definition describes how to get each term from the previous one, which is perfect for sequential processes: if you 'start with 200 and add 50 each week,' that becomes a₁ = 200, aₙ₊₁ = aₙ + 50. You need both the starting value and the rule for what comes next. The context describes a sequential process: 'start with 60 points, and each new level gives you 15 more points than the previous level.' We need a starting point—that's 60 from 'start with 60 points'—and a rule for each step. Since we get '15 more points than the previous level,' each new value is the previous value plus 15. Writing this as a recursive definition: a₁ = 60, aₙ₊₁ = aₙ + 15. To find the 5th level's points, we'd start at 60 and apply the rule four times! Choice C is correct because it includes both the initial value (a₁ = 60) and the recurrence relation (aₙ₊₁ = aₙ + 15) that matches getting 15 more points each level. Choice A has the recursive rule using multiplication: aₙ₊₁ = 15aₙ means multiply by 15, but the context says '15 more points,' which means addition, not multiplication. When you see 'more than,' that's addition—'times as many' would be multiplication! For recursive definitions, remember the two-part recipe: you MUST have both (1) the starting value(s)—look for 'starts at,' 'begins with,' 'initially'—and (2) the rule relating each term to the previous—look for 'add,' 'multiply by,' 'increases by' followed by a description. Write it as: a₁ = [starting value], aₙ₊₁ = [rule using aₙ]. If you're stuck choosing between explicit and recursive, ask: does each value only depend on which step you're at (explicit), or does each value depend on the previous value (recursive)? 'Day 5 costs $45' suggests explicit. 'Add$10 to yesterday's amount' suggests recursive. The language tells you which form fits!

Question 8

1. $C(m)=15m+25$ (correct answer)
2. $C(m)=25+15$
3. $C(m)=25m+15$
4. $C(m)=40m$

Explanation: This question tests your ability to translate a real-world situation into mathematical language—either as an explicit formula, a recursive process, or a series of calculation steps. An explicit expression gives you a direct formula to calculate the output from the input without needing any previous values: if the context says 'costs $45 per day plus$25 fee,' you can write C(d) = 45d + 25, where d is days and C is cost. Just plug in any number of days and calculate immediately! Looking at the context 'A gym membership costs $25 to join and then$15 each month,' we identify the independent variable as m (months) and dependent variable as C (total cost). The relationship is linear because there's a constant rate per month plus a fixed fee. The rate is 15 from '$15 each month,' and the initial fee is 25 from '$25 to join.' Putting this together: C(m) = 25 + 15m. This formula lets us calculate total cost for any value of months! Choice B is correct because it matches the linear structure with the joining fee added to the monthly rate times months. Choice A has the right idea but switches the numbers: the context says '$25 to join' (fixed) and '$15 each month' (rate), but this choice uses 25 as the rate and 15 as the fixed—'each' means multiply by the variable! To write explicit functions from context: (1) Underline key phrases like 'per,' 'starts at,' 'plus,' 'each'—these tell you operations and values, (2) Identify what changes (independent variable x, n, t) and what you're calculating (dependent variable C, h, P), (3) Determine function type: constant rate = linear, percent/doubling = exponential, area/motion = quadratic, (4) Extract numbers from context and plug into the right form (y = mx + b for linear, y = a·b^x for exponential, etc.). Context translation cheat sheet: 'per/each' = multiply by that rate, 'plus/in addition' = add, 'starts at' = initial value, 'doubles/triples' = multiply by 2 or 3 (exponential), 'percent' = divide by 100 for decimal. These phrases are your clues for turning words into math!

Question 9

1. $a_1=40,\ a_{n+1}=a_n+15$ (correct answer)
2. $a_1=15,\ a_{n+1}=a_n+40$
3. $a_1=40,\ a_{n+1}=15a_n$
4. $a_{n+1}=a_n+15$

Explanation: This question tests your ability to translate a real-world situation into mathematical language—either as an explicit formula, a recursive process, or a series of calculation steps. A recursive definition describes how to get each term from the previous one, which is perfect for sequential processes: if you 'start with 200 and add 50 each week,' that becomes a₁ = 200, aₙ₊₁ = aₙ + 50. You need both the starting value and the rule for what comes next. The context describes a sequential process: 'A student saves money each week. In week 1, the student saves $40. Each following week, the student saves$15 more than the previous week.' We need a starting point—that's 40 from 'In week 1, the student saves $40'—and a rule for each step. Since we 'saves$15 more than the previous week,' each new value is previous plus 15. Writing this as a recursive definition: a₁ = 40, aₙ₊₁ = aₙ + 15. To find the 5th term, we'd start at 40 and apply the rule four times! Choice A is correct because it includes both initial value and recurrence with the starting amount and the addition of 15 each week. Choice D has the recursive rule but is missing the initial value a₁ = 40. Without knowing where to start, we can't use the rule! Recursive definitions ALWAYS need both the starting value(s) and the recurrence relation. For recursive definitions, remember the two-part recipe: you MUST have both (1) the starting value(s)—look for 'starts at,' 'begins with,' 'initially'—and (2) the rule relating each term to the previous—look for 'add,' 'multiply by,' 'increases by' followed by a description. Write it as: a₁ = [starting value], aₙ₊₁ = [rule using aₙ]. If you're stuck choosing between explicit and recursive, ask: does each value only depend on which step you're at (explicit), or does each value depend on the previous value (recursive)? 'Day 5 costs $45' suggests explicit. 'Add$10 to yesterday's amount' suggests recursive. The language tells you which form fits!

Question 10

1. $P(t)=200+3t$
2. $P(t)=3(200)^t$
3. $P(t)=200\cdot 3^t$ (correct answer)
4. $P(t)=200\cdot t^3$

Explanation: This question tests your ability to translate a real-world situation into mathematical language—either as an explicit formula, a recursive process, or a series of calculation steps. The key to writing functions from context is identifying what changes (independent variable, like time or number of items) and what you're calculating (dependent variable, like cost or height), then finding the mathematical relationship between them using clues in the language. This context involves 'triples every hour,' which tells us this is an exponential function. Exponential functions have constant multiplication—the population is multiplied by the same factor each time period. The form is P(t) = a·b^t, and from the context we extract: initial population a = 200 from 'starts with 200 bacteria' and growth factor b = 3 from 'triples' = P(t) = 200·3^t. Choice C is correct because it properly identifies 200 as the initial population (when t = 0) and 3^t as the growth factor that triples the population each hour, matching the exponential structure for repeated multiplication. Choice A gives a linear function when the context describes exponential growth: 'triples every hour' indicates multiplication by 3 each hour, which means exponential growth, not adding 3 each hour. Percent growth and multiplication patterns mean exponential! If you're stuck choosing between explicit and recursive, ask: does each value only depend on which step you're at (explicit), or does each value depend on the previous value (recursive)? 'Day 5 costs $45' suggests explicit. 'Add$10 to yesterday's amount' suggests recursive. The language tells you which form fits!

Question 11

1. $H(t)=80+6t$
2. $H(t)=80-6t$ (correct answer)
3. $H(t)=6-80t$
4. $H(t)=80\cdot 6^t$

Explanation: This question tests your ability to translate a real-world situation into mathematical language—either as an explicit formula, a recursive process, or a series of calculation steps. An explicit expression gives you a direct formula to calculate the output from the input without needing any previous values: if the context says 'costs $45 per day plus$25 fee,' you can write C(d) = 45d + 25, where d is days and C is cost. Just plug in any number of days and calculate immediately! Looking at the context 'has 80 health points and loses 6 health points each minute,' we identify the independent variable as t (time in minutes) and dependent variable as H (remaining health). The relationship is linear because we have a starting amount that decreases at a constant rate. The initial health is 80 from 'has 80 health points,' and the rate is -6 per minute from 'loses 6 health points each minute.' Putting this together: H(t) = 80 - 6t. This formula lets us calculate remaining health at any time! Choice C is correct because it properly identifies the starting health (80) and subtracts the health lost over time (6t), matching the context of losing 6 points per minute. Choice A has the right numbers but uses addition instead of subtraction: the context says 'loses,' which means subtract, not add. When you see 'loses,' 'decreases,' or 'drains,' that's subtraction—the opposite of 'gains' or 'adds'! To write explicit functions from context: (1) Underline key phrases like 'per,' 'starts at,' 'plus,' 'each'—these tell you operations and values, (2) Identify what changes (independent variable x, n, t) and what you're calculating (dependent variable C, h, P), (3) Determine function type: constant rate = linear, percent/doubling = exponential, area/motion = quadratic, (4) Extract numbers from context and plug into the right form (y = mx + b for linear, y = a·b^x for exponential, etc.). Context translation cheat sheet: 'per/each' = multiply by that rate, 'plus/in addition' = add, 'starts at' = initial value, 'doubles/triples' = multiply by 2 or 3 (exponential), 'percent' = divide by 100 for decimal. These phrases are your clues for turning words into math!

Question 12

1. $C(m)=35m+18$
2. $C(m)=18+35m$
3. $C(m)=35+18m$ (correct answer)
4. $C(m)=35\cdot 18^m$

Explanation: This question tests your ability to translate a real-world situation into mathematical language—either as an explicit formula, a recursive process, or a series of calculation steps. An explicit expression gives you a direct formula to calculate the output from the input without needing any previous values: if the context says 'costs $45 per day plus$25 fee,' you can write C(d) = 45d + 25, where d is days and C is cost. Just plug in any number of days and calculate immediately! Looking at the context 'costs $35 to sign up plus$18 per month,' we identify the independent variable as m (number of months) and dependent variable as C(m) (total cost). The relationship is linear because we have a one-time fee plus a constant monthly rate. The sign-up fee is $35 from 'costs$35 to sign up,' and the monthly rate is $18 from '$18 per month.' Putting this together: C(m) = 35 + 18m. This formula lets us calculate the total cost after any number of months! Choice C is correct because it properly identifies $35 as the initial sign-up fee and$18m as the total monthly charges for m months, matching the structure 'initial fee + rate × time.' Choice A reverses the coefficients: it would mean $35 per month plus an$18 sign-up fee, but the context clearly states '$35 to sign up' (one-time) and '$18 per month' (recurring). Always match the numbers to their correct roles! To write explicit functions from context: (1) Underline key phrases like 'per,' 'starts at,' 'plus,' 'each'—these tell you operations and values, (2) Identify what changes (independent variable x, n, t) and what you're calculating (dependent variable C, h, P), (3) Determine function type: constant rate = linear, percent/doubling = exponential, area/motion = quadratic, (4) Extract numbers from context and plug into the right form (y = mx + b for linear, y = a·b^x for exponential, etc.).

Question 13

1. $V(t)=600-15t$ (correct answer)
2. $V(t)=600+15t$
3. $V(t)=15-600t$
4. $V(t)=600\left(\frac{1}{15}\right)^t$

Explanation: This question tests your ability to translate a real-world situation into mathematical language—either as an explicit formula, a recursive process, or a series of calculation steps. An explicit expression gives you a direct formula to calculate the output from the input without needing any previous values: if the context says 'costs $45 per day plus$25 fee,' you can write C(d) = 45d + 25, where d is days and C is cost. Just plug in any number of days and calculate immediately! Looking at the context 'tank contains 600 liters and drains at a constant rate of 15 liters per minute,' we identify the independent variable as t (time in minutes) and dependent variable as V(t) (remaining volume). The relationship is linear because we have a starting amount that decreases at a constant rate. The initial volume is 600 from 'contains 600 liters,' and the rate is -15 (negative because draining) from 'drains...15 liters per minute.' Putting this together: V(t) = 600 - 15t. This formula lets us calculate the remaining water after any time! Choice A is correct because it properly identifies 600 as the starting volume and subtracts 15t to represent the volume drained after t minutes, matching the decreasing linear relationship. Choice B has the right numbers but adds instead of subtracts: the context says 'drains,' which means the volume decreases, so we subtract 15t, not add it. When you see 'drains,' 'loses,' or 'decreases,' that's subtraction! To write explicit functions from context: (1) Underline key phrases like 'per,' 'starts at,' 'plus,' 'each'—these tell you operations and values, (2) Identify what changes (independent variable x, n, t) and what you're calculating (dependent variable C, h, P), (3) Determine function type: constant rate = linear, percent/doubling = exponential, area/motion = quadratic, (4) Extract numbers from context and plug into the right form (y = mx + b for linear, y = a·b^x for exponential, etc.).

Question 14

1. $T_1 = 1$; $T_n = T_{n-1} + 2n$ for $n \geq 2$
2. $T_1 = 1$; $T_n = 2T_{n-1} + 1$ for $n \geq 2$
3. $T_1 = 3$; $T_n = T_{n-1} + 2$ for $n \geq 2$
4. $T_1 = 1$; $T_n = T_{n-1} + 2$ for $n \geq 2$ (correct answer)

Explanation: When you encounter recursive sequence problems, focus on identifying the pattern between consecutive terms and the starting value. A recursive formula needs two parts: an initial term and a rule showing how each term relates to the previous one. Let's trace through this tree-planting pattern. Row 1 has 1 tree, row 2 has 3 trees, row 3 has 5 trees. The pattern shows each row has exactly 2 more trees than the previous row: 1 → 3 (add 2) → 5 (add 2) → 7 (add 2), and so on. This means $T_1 = 1$ and $T_n = T_{n-1} + 2$ for $n \geq 2$. Let's verify: $T_2 = T_1 + 2 = 1 + 2 = 3$ ✓, and $T_3 = T_2 + 2 = 3 + 2 = 5$ ✓. Choice A incorrectly adds $2n$ instead of just 2. This would give $T_2 = 1 + 2(2) = 5$, which doesn't match our pattern where the second row has 3 trees. Choice B uses $2T_{n-1} + 1$, which doubles the previous term. This would give $T_2 = 2(1) + 1 = 3$, which works for row 2, but $T_3 = 2(3) + 1 = 7$, not the correct 5 trees. Choice C starts with $T_1 = 3$, but the problem clearly states the first row has 1 tree, not 3. The correct answer is D. Study tip: Always test your recursive formula with the given values. Calculate the first few terms to verify your formula produces the correct sequence.

Question 15

1. $C(m) = \begin{cases} 50 + 30m & \text{if } m \leq 6 \\ 50 + 25m & \text{if } m > 6 \end{cases}$
2. $C(m) = \begin{cases} 50 + 30m & \text{if } m \leq 6 \\ 230 + 25m & \text{if } m > 6 \end{cases}$
3. $C(m) = \begin{cases} 50 + 30m & \text{if } m \leq 6 \\ 230 + 25(m-6) & \text{if } m > 6 \end{cases}$ (correct answer)
4. $C(m) = \begin{cases} 50 + 30m & \text{if } m \leq 6 \\ 50 + 180 + 25(m-6) & \text{if } m > 6 \end{cases}$

Explanation: The correct answer is C. For the first 6 months, the cost is $50 +$30m. After 6 months, the total cost is $50 +$30(6) = $230. For months beyond 6, the additional cost is$25 per month for each month beyond 6, so the total cost is $230 +$25(m-6). Choice A restarts the calculation ignoring previous payments. Choice B incorrectly adds 25m for all months rather than just the additional months beyond 6. Choice D is equivalent to choice C when simplified (50 + 180 + 25(m-6) = 230 + 25(m-6)), but choice C is the more direct representation.

Question 16

1. $C(m) = 35 + 0.15m$ for all values of $m$
2. $C(m) = \begin{cases} 35 + 0.15m & \text{if } m \leq 200 \\ 75 & \text{if } m > 200 \end{cases}$ (correct answer)
3. $C(m) = \begin{cases} 35 + 0.15m & \text{if } m < 200 \\ 75 + 0.15m & \text{if } m \geq 200 \end{cases}$
4. $C(m) = \begin{cases} 35 + 0.15m & \text{if } m \leq 200 \\ 75 + 0.15(m-200) & \text{if } m > 200 \end{cases}$

Explanation: The correct answer is B. For 200 miles or fewer, the cost follows the standard formula of $35 +$0.15m. For more than 200 miles, the cost is a flat $75 regardless of mileage. Choice A ignores the flat rate condition entirely. Choice C incorrectly adds the mileage charge to the flat rate for high mileage. Choice D attempts to charge the flat rate plus additional mileage beyond 200 miles, which contradicts the 'regardless of mileage' condition in the problem.

Question 17

1. $P_0 = 500$; $P_n = P_{n-1} + 3$ for $n \geq 1$
2. $P_0 = 500$; $P_n = 3P_{n-1}$ for $n \geq 1$ (correct answer)
3. $P_0 = 3$; $P_n = 500P_{n-1}$ for $n \geq 1$
4. $P_0 = 500$; $P_n = P_{n-1} + 3P_{n-1}$ for $n \geq 1$

Explanation: The correct answer is B. The initial population is 500, so P₀ = 500. Since the population triples every 4 hours, each term is 3 times the previous term, giving us Pₙ = 3P_{n-1}. Choice A shows addition instead of multiplication for tripling. Choice C has the wrong initial value and multiplication factor reversed. Choice D represents Pₙ = P_{n-1} + 3P_{n-1} = 4P_{n-1}, which would quadruple the population rather than triple it.

Question 18

1. $W(t) = \begin{cases} 1200 - 15t & \text{if } 0 \leq t \leq 30 \\ 1200 - 25t & \text{if } t > 30 \end{cases}$
2. $W(t) = \begin{cases} 1200 - 15t & \text{if } 0 \leq t \leq 30 \\ 1200 - 25(t-15) & \text{if } t > 30 \end{cases}$
3. $W(t) = \begin{cases} 1200 - 15t & \text{if } 0 \leq t \leq 30 \\ 1200 - 15(30) - 25t & \text{if } t > 30 \end{cases}$
4. $W(t) = \begin{cases} 1200 - 15t & \text{if } 0 \leq t \leq 30 \\ 750 - 25(t-30) & \text{if } t > 30 \end{cases}$ (correct answer)

Explanation: When you encounter piecewise functions modeling real-world scenarios with changing rates, you need to carefully track what happens at each stage and ensure continuity between the pieces. For the first 30 minutes, water drains at 15 gallons per minute from the initial 1200 gallons, so $W(t) = 1200 - 15t$ for $0 \leq t \leq 30$. At $t = 30$, this gives us $W(30) = 1200 - 15(30) = 750$ gallons remaining. After 30 minutes, the rate changes to 25 gallons per minute. The key insight is that you start this second phase with 750 gallons (not the original 1200), and you only count the additional time beyond 30 minutes at the new rate. So for $t > 30$: $W(t) = 750 - 25(t-30)$. Choice A incorrectly restarts from 1200 gallons at $t = 30$, ignoring the water already drained in the first phase. Choice B has the wrong rate in the second piece and an illogical $(t-15)$ term. Choice C attempts to account for the first 30 minutes by subtracting $15(30) = 450$, but then incorrectly subtracts $25t$ for the entire time $t$, rather than just the time after 30 minutes. Only choice D correctly shows that after 30 minutes, you have 750 gallons left and drain at 25 gallons per minute for each additional minute beyond the first 30. Study tip: In piecewise rate problems, always check that your function pieces connect properly at the boundary points—the amount at the end of one phase must equal the starting amount for the next phase.

Question 19

1. $h(n) = 64 \left(\frac{3}{4}\right)^n$ (correct answer)
2. $h(n) = 64 \left(\frac{3}{4}\right)^{n-1}$
3. $h(n) = 64 - n \cdot \frac{3}{4}$
4. $h(n) = 64 \left(\frac{4}{3}\right)^n$

Explanation: The correct answer is A. Before any bounces (n = 0), the ball is at 64 feet. After 1 bounce (n = 1), the height is 64 × (3/4) = 48 feet. After 2 bounces (n = 2), the height is 64 × (3/4)² = 36 feet. The pattern shows h(n) = 64(3/4)ⁿ. Choice B would give h(1) = 64(3/4)⁰ = 64, meaning the ball doesn't lose height after the first bounce. Choice C represents linear decay rather than exponential decay. Choice D uses the reciprocal ratio 4/3, which would make the ball gain height with each bounce.

Question 20

1. $C(n) = 12.99n + 2.50$
2. $C(n) = 2.50n + 12.99$ (correct answer)
3. $C(n) = 12.99 + 2.50 + n$
4. $C(n) = (12.99 + 2.50) \cdot n$

Explanation: The correct answer is B. The monthly subscription fee of $12.99 is a fixed cost that doesn't change regardless of the number of rentals, so it's the constant term. The$2.50 per rental is the variable cost that depends on n, making it the coefficient of n. Therefore, C(n) = 2.50n + 12.99. Choice A incorrectly treats the subscription fee as the coefficient of n. Choice C adds n as a separate term rather than as a coefficient. Choice D multiplies the entire sum by n, which would mean both costs depend on the number of rentals.