Functions, Patterns and Sequences: Lesson Test

Answer: D

Correct answer: 17. Substitute 4 for x:

• A: This multiplies 3, 4 and 5 together, treating the plus sign as a times sign.
• B: This adds 5 to the input before multiplying, computing 3(4 + 5) instead of 3(4) + 5.
• C: This is only 3(4); the constant 5 was never added.
• D (correct): Replacing x with 4 gives 3(4) + 5 = 12 + 5 = 17.

Answer: A

Correct answer: {(1, 3), (2, 5), (1, 7)}. A relation fails to be a function only when one input has two different outputs. In {(1, 3), (2, 5), (1, 7)}, the input 1 goes to both 3 and 7.

• A (correct): The input 1 is paired with two different outputs, 3 and 7, so this relation breaks the one-output rule.
• B: Every input appears once with a single output, so this is a function.
• C: The pair (5, 1) is just listed twice; input 5 still has only one output, so this is a function.
• D: Several inputs sharing the output 4 is allowed; each input still has only one output.

Answer: A

Correct answer: A circle centered at the origin. A graph is a function only if no vertical line touches it more than once. A vertical line through the center of a circle hits the top and the bottom, so the circle fails.

• A (correct): A vertical line through the middle of a circle crosses it twice, so one input has two outputs.
• B: Each vertical line meets an upward parabola only once, so it passes the test.
• C: Every input has the single output 3; a vertical line touches it once, so it passes.
• D: Any vertical line meets a slanted line exactly once, so it passes the test.

Answer: B

Correct answer: {1, 5, 9}. The range is every second coordinate that appears: 5, 9, 5 and 1. Listing each value once gives {1, 5, 9}.

• A: This keeps only the smallest and largest outputs and leaves out 5, which is also produced.
• B (correct): The range is the set of outputs produced: 5, 9, 5 and 1, which is {1, 5, 9} once the repeat is listed once.
• C: These are the inputs, so this is the domain, not the range.
• D: This mixes inputs and outputs together; the range uses only the outputs.

Answer: B

Correct answer: 162. Divide consecutive terms:

so the common ratio is 3. The next term is 54 times 3 = 162.

• A: This adds 18, the previous term, rather than multiplying by the common ratio.
• B (correct): Each term is 3 times the one before (r = 54 / 18 = 3), so the next term is 54 times 3 = 162.
• C: This adds 36, the last gap, treating a geometric pattern as if the gaps were steady.
• D: This doubles 54, using a ratio of 2 instead of the actual ratio of 3.

Answer: D

Correct answer: 7. Set the rule equal to the output:

Add 7 to get 4x = 28, then divide by 4 to get x = 7.

• A: This is f(21), the output when 21 is used as the input, not the input that gives 21.
• B: This stops at 21 - 7 and never divides by 4, and the sign of the 7 is also handled wrongly.
• C: This subtracts 7 from 21 instead of adding it, solving 4x = 14.
• D (correct): Setting 4x - 7 = 21 gives 4x = 28, so x = 7; check: 4(7) - 7 = 21.

Answer: B

Correct answer: 12. Substitute -2 in parentheses: $(-2{)}^2-3(-2)+2$.

That is

• A: This squares without parentheses, writing $-2^2=-4$, then gets -4 + 6 + 2.
• B (correct): With parentheses, $(-2{)}^2-3(-2)+2=4+6+2=12$.
• C: This takes -3 times -2 as -6 instead of +6, giving 4 - 6 + 2.
• D: This makes both sign errors: -4 for the square and -6 for the middle term.

Answer: C

Correct answer: x ≥ 3. A square root needs a nonnegative inside:

Adding 6 gives

The value 3 itself works because the square root of 0 is 0.

• A: This flips the sign of the 3 when solving 2x - 6 = 0.
• B: This wrongly excludes x = 3, but the square root of 0 is 0, which is allowed.
• C (correct): The inside 2x - 6 must be at least 0, so 2x is at least 6 and x is at least 3.
• D: This sets x itself at least 6 and forgets to divide by the 2 in 2x.

Answer: C

Correct answer: 172. The common difference is

• A: This multiplies the first term 4 by 25, mixing up the roles of the numbers.
• B: This uses n instead of n - 1, adding one extra step of 7.
• C (correct): With d = 7, the 25th term is 4 + (25 - 1)(7) = 4 + 168 = 172.
• D: This multiplies 25 by 7 and ignores the first term of 4.

Answer: D

Correct answer:

Divide each output change by its input change:

so the slope is 3. Then

gives b = 2, so

• A: This takes the first output as the starting value, but 5 is the output at x = 1, not at x = 0.
• B: This matches only the first row; at x = 3 it gives 9, not 11.
• C: This uses the output change of 6 as the slope without dividing by the input change of 2; at x = 3 it gives 17, not 11.
• D (correct): The rate is 6 / 2 = 3 and 12 / 4 = 3, and 3(1) + 2 = 5, so this rule matches every row.

Answer: C

Correct answer: 19. Apply g first:

Then apply f to that result:

• A: This adds f(3) = 5 and g(3) = 10 instead of composing the functions.
• B: This is g(f(3)) = g(5) = 26; it applies f first, which reverses the required order.
• C (correct): Working inside out, g(3) = 10, and then f(10) = 2(10) - 1 = 19.
• D: This multiplies f(3) = 5 by g(3) = 10 instead of feeding one result into the other function.

Answer: C

Correct answer: 5. The boundary input 3 falls under the condition x ≥ 3, so use

• A: This uses the first rule, 2(3) + 1, but that rule covers only inputs less than 3.
• B: This picks the correct rule but forgets to subtract 4 after squaring.
• C (correct): Since 3 satisfies x ≥ 3, the second rule applies: 9 - 4 = 5.
• D: This adds the results of both rules; only one rule applies to any single input.

Answer: A

Correct answer: $160. Substitute m = 8 into

15(8) = 120, and

so the cost is $160.

• A (correct): C(8) = 15(8) + 40 = 120 + 40 = 160 dollars.
• B: This counts only the monthly charges and leaves out the one-time $40 fee.
• C: This swaps the roles, charging $40 per month plus a $15 fee: 40(8) + 15.
• D: This charges the $40 fee every month, computing 8 times $55.

Answer: C

Correct answer: 384. The ratio is

• A: This uses $2^8$, raising the ratio to n instead of n - 1.
• B: This multiplies the first term by 8, treating the position number as the ratio.
• C (correct): With r = 2, the 8th term is $3\cdot 2^7=3\cdot 128=384$.
• D: This uses $2^6$, stopping one multiplication short of the 8th term.

Answer: A

Correct answer: 820. First find the last term:

• A (correct): The 20th term is 3 + 19(4) = 79, and the sum is 20(3 + 79) / 2 = 820.
• B: This finds the 20th term as 3 + 20(4) = 83 by using n instead of n - 1, giving 20(86) / 2.
• C: This is only the 20th term, not the sum of all 20 terms.
• D: This multiplies 20 by 82 but never divides by 2.

Answer: B

Correct answer: The 17th term. Set up

and n = 17. Check:

• A: This uses the total drop of 112 as the position instead of dividing it by 7.
• B (correct): Solving 120 - 7(n - 1) = 8 gives n - 1 = 16, so n = 17.
• C: This adds one step too many; the 18th term is 120 - 7(17) = 1.
• D: This stops at n - 1 = 16 and forgets to add 1 to get n.

Answer: D

Correct answer: $7,680. Each year the value is multiplied by 0.8. After 3 years: 15,000 times 0.8 = 12,000, then 9,600, then 7,680 dollars.

• A: This is the value after only 2 years, $15000\cdot 0.8^2$.
• B: This subtracts 20% of the original $15,000 three times, treating the decay as a fixed amount instead of a percent of the current value.
• C: This multiplies by 0.2 each year, keeping the 20% lost instead of the 80% kept.
• D (correct): Keeping 80% each year gives $15000\cdot 0.8^3=7680$ dollars.

Answer: A

Correct answer: 77. Apply the rule step by step:

• A (correct): Building in order gives 2, 7, 17, 37, 77, so the 5th term is 77.
• B: This is the 4th term; one more application of the rule is needed.
• C: This uses the shortcut $2\cdot 2^4+3$, but the +3 is added at every step, so terms must be built one at a time.
• D: This is the 6th term; the rule was applied one time too many.

Answer: B

Correct answer: 104. The first gaps 3, 5, 7, 9 have constant second gaps of 2, so the pattern is quadratic. Comparing with 1, 4, 9, 16, 25 shows each term is $n^2+4$, so the 10th term is

• A: This uses $9^2+4$, which is the 9th term, by plugging in n - 1 instead of n.
• B (correct): The gaps 3, 5, 7, 9 rise by 2, so the rule is $n^2+4$, and $10^2+4=104$.
• C: This treats the pattern as arithmetic with d = 3, but the gaps are not constant.
• D: This is only the 6th term, the next one in the list.

Answer: D

Correct answer: 20. The even positions form their own list, 8, 11, 14, ..., rising by 3. The 10th term is the 5th even-position term:

• A: This is the 8th term; the count stopped two positions early.
• B: This is the 9th term; one more step of adding 5 is needed.
• C: This is the 12th term; two extra steps were counted.
• D (correct): Continuing the steps gives 12, 17, 15, 20, so the 10th term is 20.

Answer: A

Correct answer: 17. Work inside out. First

Then feed 6 into g:

• A (correct): f(−2) = 4 + 2 = 6, and g(6) = 18 − 1 = 17.
• B: This applies g first and f second: g(−2) = −7 and f(−7) = 51, which is f(g(−2)).
• C: This squares −2 as −4, so f(−2) becomes −2 and g(−2) = −7; a negative squared is positive.
• D: This is only the inside value f(−2); it still has to be put into g.

Answer: A

Correct answer: 9. Replace x with 3/4:

• A (correct): Eight times 3/4 is 6, and adding 3 gives 9.
• B: This multiplies 8 by 3/4 but forgets to add the 3.
• C: This substitutes only the numerator 3 for x, giving 8(3) + 3.
• D: This substitutes only the denominator 4 for x, giving 8(4) + 3.

Answer: B

Correct answer: 6. Set the rule equal to the output and solve:

Subtract 7:

Divide by −3: x = 6. Check:

• A: This loses the sign when dividing −18 by −3; a negative divided by a negative is positive.
• B (correct): Solving 7 − 3x = −11 gives −3x = −18, so x = 6.
• C: This evaluates f(−11) = 7 + 33 instead of finding the input that produces −11.
• D: This adds 7 to −11 instead of subtracting it, giving −3x = −4.

Answer: A

Correct answer: x ≥ 1, except x = 3. Two restrictions apply. The square root needs

so x ≥ 1. The denominator cannot be zero, so x ≠ 3. The domain is all x ≥ 1 except x = 3.

• A (correct): This keeps the square root nonnegative and avoids a zero denominator at x = 3.
• B: This handles the square root but allows x = 3, which makes the denominator zero.
• C: This wrongly throws out inputs such as 2, where the root and the denominator are both fine.
• D: This bans only x = 1 and x = 3; x = 1 is allowed, and inputs below 1 make the root impossible.

Answer: D

Correct answer: {(4, 1), (5, 1), (6, 1)}. A relation is a function when no input is paired with two different outputs. In {(4, 1), (5, 1), (6, 1)} each input appears once; sharing the output 1 is allowed.

• A: The input 1 is paired with both 4 and 5, so this is not a function.
• B: The input 3 is paired with both 2 and −2, so this is not a function.
• C: The input 0 is paired with both 1 and 5, so this is not a function.
• D (correct): Each input has exactly one output; different inputs may share the same output.

Answer: D

Correct answer: {−1, 2, 5}. The domain is the set of inputs, the first numbers in the pairs: −1, 2, 5 and 2. A repeated value is listed once in a set, so the domain is {−1, 2, 5}. The repeated identical pair (2, 4) does not stop this from being a function.

• A: This is the range, the set of outputs, not the inputs.
• B: This mixes inputs and outputs together; the domain uses only the first coordinates.
• C: This drops the input 2 because its pair appears twice; a repeated pair still contributes its input once.
• D (correct): These are the distinct first coordinates of the pairs.

Answer: A

Correct answer: −1. Replace x with 5:

• A (correct): Substituting 5 gives 9 − 10 = −1.
• B: This drops the negative sign; 9 − 10 is −1, not 1.
• C: This adds 2(5) to 9 instead of subtracting it.
• D: This subtracts first, (9 − 2) × 5, ignoring that multiplication comes before subtraction.

Answer: D

Correct answer:

The inputs do not rise by 1, so divide each output change by its input change: from x = 2 to x = 5 the output falls 6 over 3 steps, a rate of −2, and from 5 to 11 it falls 12 over 6, again −2. Then

gives b = 10, so

• A: This uses the output change of −6 as the slope without dividing by the input change of 3; it fits only the first point.
• B: This loses the sign of the slope; the outputs fall as x rises, so the slope is negative.
• C: This treats the first output, 6, as the starting value, but 6 is the output at x = 2, not at x = 0.
• D (correct): It gives 6, 0 and −12 for x = 2, 5 and 11.

Answer: D

Correct answer: 11. Each term is 5 less than the one before, so the common difference is −5. The next term is

• A: This adds 5 instead of subtracting it; the terms are falling.
• B: This skips ahead two steps, 16 − 10, instead of one.
• C: This subtracts 4 instead of the actual difference of 5.
• D (correct): Subtracting the common difference of 5 from 16 gives 11.

Answer: C

Correct answer: 46. Here

and d = 1.5. Use

• A: This uses n = 30 steps instead of n − 1 = 29, overshooting by one difference of 1.5.
• B: This multiplies 30 by 1.5 and forgets to start from the first term.
• C (correct): Twenty-nine steps of 1.5 added to 2.5 give 46.
• D: This is only the total of the 29 steps; the first term 2.5 must be added.

Answer: C

Correct answer: The 20th term. Set up

Subtract 7:

and n = 20. Check:

• A: This stops at n − 1 = 19 and forgets to add 1 to get the term number.
• B: This adds one step too many; the 21st term is 7 + 20(6) = 127.
• C (correct): Solving 7 + 6(n − 1) = 121 gives n − 1 = 19, so n = 20.
• D: This divides 121 by the first term 7 instead of using the common difference.

Answer: B

Correct answer: 1,050. The rows form an arithmetic sequence with

and d = 2. The last row has

seats. The total is

seats.

• A: This finds the last row with 25 steps instead of 24, giving 68 seats, so the total is too large.
• B (correct): The last row has 66 seats, and 25 rows averaging (18 + 66)/2 = 42 seats give 1,050.
• C: This multiplies 25 by the sum of the first and last rows but forgets to divide by 2.
• D: This is the number of seats in the last row, not the total for all rows.

Answer: C

Correct answer: −486. Find the ratio:

An odd power of a negative number is negative.

• A: This drops the sign; −3 raised to the odd power 5 is negative.
• B: This uses the exponent n = 6 instead of n − 1 = 5, which gives 2(−3)^6 = 1,458, the 7th term.
• C (correct): Using $a_6=2(-3{)}^5$ gives 2 times −243, which is −486.
• D: This treats the pattern as arithmetic with difference −8; the terms are multiplied by −3, not shifted by a fixed amount.

Answer: B

Correct answer: 2,700. A 25% loss leaves 75% each year, so multiply by 0.75 three times: 6,400 → 4,800 → 3,600 → 2,700. In formula form,

• A: This subtracts 25% of the original 6,400 (1,600) each year, treating the decay as a fixed amount rather than a fixed percent.
• B (correct): Multiplying by 0.75 for each of 3 years gives 6,400 × 0.421875 = 2,700.
• C: This is the population after 2 years; one more year of decay is needed.
• D: This is the population after only 1 year.

Answer: C

Correct answer: 1/2. Divide any term by the one before it:

Each term is half the previous one, so

a ratio between 0 and 1 because the terms shrink.

• A: This divides the earlier term by the later one; the ratio is later ÷ earlier.
• B: This is the difference between the first two terms, which would describe an arithmetic sequence.
• C (correct): Each term divided by the previous term gives 1/2.
• D: This compares terms two places apart, such as 192 ÷ 768, instead of consecutive terms.

Answer: B

Correct answer: 80. The gaps 3, 5, 7, 9 rise by 2 each time, so the pattern is quadratic. Testing $n^2-1$:

The 9th term is

• A: This is only the next term, the 6th; the question asks for the 9th.
• B (correct): The rule is n² − 1, and 81 − 1 = 80.
• C: This is the 8th term, 8² − 1, one position short.
• D: This is the 10th term, 10² − 1, one position too far.

Answer: A

Correct answer:

The fixed fee is the starting value and the hourly charge is the rate, so

For example, 2 hours costs

• A (correct): The rate 48 multiplies the hours, and the one-time fee 65 is added once.
• B: This swaps the roles, charging the service fee every hour and the hourly rate only once.
• C: This adds the fee to the rate, so the $65 fee would be charged again for every hour.
• D: This multiplies the fee by the hourly rate, making the fixed fee cost 48 × 65.

Answer: B

Correct answer: 43. Build the terms in order:

• A: This is the third term; one more step is needed to reach the fourth.
• B (correct): Repeating "multiply by 4, then subtract 1" three times from 1 gives 3, 11, 43.
• C: This is the fifth term, one step too many.
• D: This plugs n = 4 into 4n − 1 instead of using the previous term.

Answer: C

Correct answer: 15. The odd positions count down by 5 (50, 45, 40, ...) and the even positions count up by 3 (3, 6, 9, ...). The 10th term is the 5th even-position term: 3, 6, 9, 12, 15. So the 10th term is 15.

• A: This is the 4th even-position term, which is the 8th term of the pattern.
• B: This is the 9th term, the 5th value in 50, 45, 40, ...
• C (correct): The 10th position is the 5th term of 3, 6, 9, ..., which is 15.
• D: This continues the falling sequence one step past 40, which is the 7th term.

Answer: A

Correct answer: 15. Both boundary values belong to the middle rule.

and since 5 > 2,

The sum is

• A (correct): Using the middle rule at both boundaries gives 1 + 7, and t(5) = 7, for a total of 15.
• B: This uses the first rule at x = −1, giving 0; the middle rule includes −1 because of the ≤ sign.
• C: This uses the third rule at x = 2, giving 10; the third rule applies only when x is greater than 2.
• D: This applies the outer rules at both boundary points, giving 0 + 10 + 7.

Answer: B

Correct answer: Each input is paired with exactly one output. A function assigns each input exactly one output. Several inputs may share the same output, but no input may have two outputs.

• A: This reverses the rule; different inputs are allowed to share an output.
• B (correct): This is the defining rule of a function.
• C: Sharing outputs is allowed, as in {(1, 5), (2, 5)}.
• D: Inputs can be negative or zero; the rule concerns how many outputs each input has.

Answer: D

Correct answer: 7. Work inside out:

Setting

gives k = 7.

• A: This applies f first and g second: f(5) − 3 = 7 + k = 11 gives k = 4, which solves g(f(5)) = 11.
• B: This skips g and evaluates f(5) = 10 + k directly.
• C: This feeds g(5) = 2 into f but forgets to double it, so 2 + k = 11.
• D (correct): g(5) = 2 and f(2) = 4 + k, so k = 11 − 4 = 7.

Answer: C

Correct answer: 15 hours. Set the rule equal to the output:

Subtract 840:

Divide by −35: h = 15 hours. Check:

• A: This divides 315 by 35, treating the amount remaining as the amount drained.
• B: This divides 840 by 35, which is the time to empty the tank completely.
• C (correct): Solving 840 − 35h = 315 gives 35h = 525, so h = 15.
• D: This adds 315 to 840 instead of subtracting, giving 1,155 ÷ 35.

Answer: D

Correct answer: −3. Substitute with parentheses:

• A: This treats both parts as positive, 9 + 12, dropping the sign on 4(−3).
• B: This squares −3 as −9 by leaving out the parentheses, so −9 − 12 = −21.
• C: This drops the final sign; 9 − 12 is −3, not 3.
• D (correct): The square of −3 is 9, and 4 times −3 is −12, so the total is −3.