High-school Functions objective evidence

Read domains and ranges across representations — Algebra II and precalculus: functions and structure

• Which listed relation fails to define a function of its first coordinate?Automatically checked response; reasoning still matters
• A function is defined only by the pairs (−2,4), (0,1), (3,4), (5,9). Enter its range as a finite set.Automatically checked response; reasoning still matters
• Find the domain and range of f(x)=2+√(5−x) over the reals. Explain why every value in your claimed range is attained.Written reasoning: review required

Choose the correct branch at a boundary — Algebra II and precalculus: functions and structure

• Let f(x)=2x+1 for x<3 and f(x)=10−x for x≥3. Find f(3).Automatically checked response; reasoning still matters
• A fictional delivery service charges $8 for mass m with 0<m≤2 kg, then $3 for each additional kg, charged proportionally, for 2<m≤6. Write the complete rule and find the charges for 2 and 4.5 kg. State its domain.Written reasoning: review required

Connect functions through intermediate values — Algebra II and precalculus: functions and structure

• A fictional sensor maps raw count r to T(r)=r/4−5 degrees Celsius. A display maps C to F(C)=9C/5+32 degrees Fahrenheit. What Fahrenheit value corresponds to r=60?Automatically checked response; reasoning still matters

Translate recursive and explicit sequences — Functions in graphs, measurements and growth decisions

• Supplied Graph D plots sequence values 2,4,8,16 at indices 1,2,3,4. Explain why the graph has separate points and whether a value at index 2.5 is defined.Written reasoning: review required
• Translate a_n=30−3(n−1), n≥1, into a recursive rule; then derive the explicit rule back from that recursion.Written reasoning: review required

Compare functions across representations — Functions in graphs, measurements and growth decisions

• A signed account balance starts at −4 dollars, rises steadily to 2 dollars after 3 days, remains there for 2 days, then falls steadily to −1 dollar at day 8. Construct a complete piecewise-linear graph description and identify zeros, maximum and decreasing interval.Written reasoning: review required

Combine meaningful functions into a quadratic decision — Functions in graphs, measurements and growth decisions

• A proper rectangle has perimeter 28 m and one side x m. Construct its area function and graph description, using factored and vertex forms. State the contextual domain, zero-area interpretation and maximum.Written reasoning: review required
• Using the supplied revenue/profit graph for n(p)=120−4p, explain why price 15 maximizes revenue but not profit. Then critique continuing that model to p=40 and calling the resulting negative sales a valid prediction. What evidence would a real pricing decision require?Written reasoning: review required

Fit a periodic model and test its assumptions — Trigonometry, polar geometry and precalculus representations

• A fictional wheel has radius5 m and centre6 m above ground, completes a revolution every20 seconds, and begins with a rider at the bottom. Write a height model and find heights at t=0,5,10.Written reasoning: review required

Construct polynomial and rational graphs — Functions in graphs, measurements and growth decisions

• Construct Q(x)=−(x+2)(x−1)² as a graph or a complete coordinate-and-branch sketch description. Include roots and their multiplicities, crossing/touching behavior, y-intercept, signs and both ends. Supply at least one additional checked landmark.Written reasoning: review required
• A student draws x⁴−5x²+4 with opposite end directions because it has several roots. Factor it, specify all roots and sign intervals, then repair the ends and explain which algebraic feature controls them.Written reasoning: review required

Test even and odd symmetry with domains — Functions in graphs, measurements and growth decisions

• Classify supplied Graph P and Graph Q using their complete piecewise-linear definitions, not only a few points. Explain the symmetry of each segment.Written reasoning: review required

Graph ceiling steps and discrete decisions — Functions in graphs, measurements and growth decisions

• A machine processes only whole batches n=0,1,2,... and cost is 7n dollars. Should its graph include the whole line y=7x? Explain.Written reasoning: review required

Combine meaningful functions into a quadratic decision — Functions in graphs, measurements and growth decisions

• A proper rectangle has perimeter 28 m and one side x m. Construct its area function and graph description, using factored and vertex forms. State the contextual domain, zero-area interpretation and maximum.Written reasoning: review required

Read rates from uneven tables and graph scales — Functions in graphs, measurements and growth decisions

• A tank table gives (time minutes, volume liters): (0,12),(3,21),(8,31). What is the average rate from 3 to 8 minutes?Automatically checked response; reasoning still matters
• In supplied Graph R, read values at 4 and 10 minutes and calculate the average rate between them in liters per minute.Automatically checked response; reasoning still matters
• On a plotted graph the vertical reading uncertainty is about ±1 L at each endpoint, and endpoint times 2 and 7 minutes are exact. The estimated volumes are 14 and 24 L. Give a compatible interval for the secant rate.Written reasoning: review required

Compare functions across representations — Functions in graphs, measurements and growth decisions

• A signed account balance starts at −4 dollars, rises steadily to 2 dollars after 3 days, remains there for 2 days, then falls steadily to −1 dollar at day 8. Construct a complete piecewise-linear graph description and identify zeros, maximum and decreasing interval.Written reasoning: review required

Combine meaningful functions into a quadratic decision — Functions in graphs, measurements and growth decisions

• A proper rectangle has perimeter 28 m and one side x m. Construct its area function and graph description, using factored and vertex forms. State the contextual domain, zero-area interpretation and maximum.Written reasoning: review required

Construct cube-root and absolute-value graphs — Functions in graphs, measurements and growth decisions

• Construct y=cube_root(x+1)−2 using at least five exact points, with domain, range and end behavior.Written reasoning: review required
• Construct y=3|x−2|−6. Give vertex, intercepts, branch slopes, domain and range.Written reasoning: review required
• Construct y=3−2√(x+1) over the reals as a labeled graph or complete coordinate-and-curve description. Give domain, range, included endpoint, both intercepts and at least four exact landmarks. Explain why squaring the equation without a sign restriction produces an extra branch.Written reasoning: review required

Graph ceiling steps and discrete decisions — Functions in graphs, measurements and growth decisions

• Construct C(t)=5 ceiling(t/2) for t≥0 with C(0)=0. Describe the graph through t=8, including every endpoint convention.Written reasoning: review required

Read domains and ranges across representations — Algebra II and precalculus: functions and structure

• Find the domain and range of f(x)=2+√(5−x) over the reals. Explain why every value in your claimed range is attained.Written reasoning: review required

Choose the correct branch at a boundary — Algebra II and precalculus: functions and structure

• Let f(x)=x+1 for x<0 and f(x)=x+4 for x≥0. Sketch or describe both branches with endpoint markers. Can changing only f(0) make the graph continuous? Explain.Written reasoning: review required

Construct polynomial and rational graphs — Functions in graphs, measurements and growth decisions

• Construct Q(x)=−(x+2)(x−1)² as a graph or a complete coordinate-and-branch sketch description. Include roots and their multiplicities, crossing/touching behavior, y-intercept, signs and both ends. Supply at least one additional checked landmark.Written reasoning: review required
• A student draws x⁴−5x²+4 with opposite end directions because it has several roots. Factor it, specify all roots and sign intervals, then repair the ends and explain which algebraic feature controls them.Written reasoning: review required

Construct polynomial and rational graphs — Functions in graphs, measurements and growth decisions

• Construct F(x)=(x²−4)/(x²+x−6). Show original domain, hole, asymptotes, intercepts and branch behavior; explain every cancellation restriction.Written reasoning: review required
• Use supplied Graph P and Graph R. A learner says both inputs −1 and 2 are x-intercepts of both graphs because the corresponding factors appear in the formulas. Audit the claim and use the plotted behavior to distinguish the four cases.Written reasoning: review required

Graph transformed exponentials and logarithms — Functions in graphs, measurements and growth decisions

• Construct y=(1/2)^(x+1)−3 with three exact landmarks, asymptote, intercepts, domain, range and end behavior.Written reasoning: review required
• Construct y=log_(1/2)(x−1)+2 with exact landmarks, asymptote, domain, range and intercepts.Written reasoning: review required

Build sine and cosine graphs from rotation — Trigonometry, polar geometry and precalculus representations

• Give five quarter-period landmarks for y=2cos(πt/2) over one period starting at t=0. Include the period and range.Written reasoning: review required
• Explain why sin(2x−π/3) shifts sin(2x) right by π/6, not π/3. Give one landmark that checks the claim.Written reasoning: review required

Combine meaningful functions into a quadratic decision — Functions in graphs, measurements and growth decisions

• A proper rectangle has perimeter 28 m and one side x m. Construct its area function and graph description, using factored and vertex forms. State the contextual domain, zero-area interpretation and maximum.Written reasoning: review required
• For the new event n(p)=150−5p on [0,30], with fixed cost 100 dollars and 4 dollars per ticket, construct revenue, cost and profit functions with units. Compare the revenue-maximizing and profit-maximizing prices and their profits.Written reasoning: review required

Fit growth models and change time units — Functions in graphs, measurements and growth decisions

• An ideal exponential model multiplies by 1.21 per year. What is its equivalent factor per half-year?Automatically checked response; reasoning still matters
• A quantity grows by 8 percent each month. Give its model from initial value 50, its exact yearly factor and approximate yearly percentage; explain why 96 percent is not exact.Written reasoning: review required
• Rewrite P(t)=20·2^(t/3), t in days, using h in hours and w in weeks. Verify the same prediction at six days.Written reasoning: review required

Compare functions across representations — Functions in graphs, measurements and growth decisions

• Compare the largest outputs of supplied Graph A and B(x)=7−2(x−1)² on [0,4]. Give where each occurs.Written reasoning: review required
• A third function is sampled only at (0,1),(2,7),(4,3). Can its maximum on [0,4] be certified as 7? Explain with two possible continuous extensions.Written reasoning: review required
• A model gives height h(t)=t(6−t) meters for 0≤t≤6 seconds. Produce a graph or coordinate/curve description, rewrite in vertex form and explain both forms and the domain.Written reasoning: review required

Translate recursive and explicit sequences — Functions in graphs, measurements and growth decisions

• A fund begins with 200 dollars at index 0. Model adding 20 dollars after each month versus multiplying the balance by 1.1 after each month. Give both recursive and explicit models, and compare at month 3.Written reasoning: review required

Choose the correct branch at a boundary — Algebra II and precalculus: functions and structure

• A fictional delivery service charges $8 for mass m with 0<m≤2 kg, then $3 for each additional kg, charged proportionally, for 2<m≤6. Write the complete rule and find the charges for 2 and 4.5 kg. State its domain.Written reasoning: review required

Combine meaningful functions into a quadratic decision — Functions in graphs, measurements and growth decisions

• For the new event n(p)=150−5p on [0,30], with fixed cost 100 dollars and 4 dollars per ticket, construct revenue, cost and profit functions with units. Compare the revenue-maximizing and profit-maximizing prices and their profits.Written reasoning: review required

Connect functions through intermediate values — Algebra II and precalculus: functions and structure

• A fictional sensor maps raw count r to T(r)=r/4−5 degrees Celsius. A display maps C to F(C)=9C/5+32 degrees Fahrenheit. What Fahrenheit value corresponds to r=60?Automatically checked response; reasoning still matters
• A fictional seller applies a 20% discount D(p)=0.8p and a $15 coupon K(p)=p−15. Compare D(K(100)) and K(D(100)); state why order matters and a reasonable common input domain if negative charges are forbidden.Written reasoning: review required
• Let f(u)=1/u and g(x)=1/(x−2). Simplify f(g(x)) and give its actual domain. Explain why the formula alone might mislead.Written reasoning: review required

Translate recursive and explicit sequences — Functions in graphs, measurements and growth decisions

• Translate a_n=30−3(n−1), n≥1, into a recursive rule; then derive the explicit rule back from that recursion.Written reasoning: review required
• A geometric sequence begins at index 3 with c_3=5 and doubles per step. Give recursive and explicit rules and compute c_7.Written reasoning: review required
• A fund begins with 200 dollars at index 0. Model adding 20 dollars after each month versus multiplying the balance by 1.1 after each month. Give both recursive and explicit models, and compare at month 3.Written reasoning: review required

Recover transformations from supplied graphs — Functions in graphs, measurements and growth decisions

• Recover the full rule for Graph G from its supplied graph, and check a point not used to find the scale.Written reasoning: review required
• A parent graph f is defined only on [−2,3]. For g(x)=−3f(x−4)+2, give the new domain and the image of parent point (1,−2).Written reasoning: review required

Test even and odd symmetry with domains — Functions in graphs, measurements and growth decisions

• Classify supplied Graph P and Graph Q using their complete piecewise-linear definitions, not only a few points. Explain the symmetry of each segment.Written reasoning: review required
• On the real line, classify x⁴−2x²+5, x³−4x and x²+x. Justify algebraically.Written reasoning: review required
• Classify f(x)=x² with domain [−1,3] as even, odd or neither under the standard symmetry definition.Written reasoning: review required

Reverse a function on a suitable domain — Algebra II and precalculus: functions and structure

• For f(x)=5x−7 on the reals, find f⁻¹(18).Automatically checked response; reasoning still matters
• A learner says x² on all real numbers has inverse √x because squaring √x returns x. Explain the missing condition and give two different valid restrictions with their inverses.Written reasoning: review required

Reverse a function on a suitable domain — Algebra II and precalculus: functions and structure

• Find the inverse of f(x)=(3x−2)/5 on the reals and verify both compositions symbolically.Written reasoning: review required

Read inverses without a formula — Functions in graphs, measurements and growth decisions

• The finite table T has input-output pairs (−3,8),(1,−2),(4,5),(9,11). Find T^(-1)(5).Automatically checked response; reasoning still matters
• Using supplied Graph I, find I^(-1)(4).Automatically checked response; reasoning still matters
• Construct the inverse of supplied Graph I as a coordinate/segment description. Give its domain and range.Written reasoning: review required
• For the finite table T, list the domain of T inverse and explain why T inverse at 6 cannot be inferred by interpolation.Written reasoning: review required

Read inverses without a formula — Functions in graphs, measurements and growth decisions

• The full parabola y=(x−1)² has outputs 4 at inputs −1 and 3. Explain why its inverse relation is not a function and give a restriction that produces an inverse, including the inverse value at 4.Written reasoning: review required

Solve logarithmic equations on their real domains — Algebra II and precalculus: functions and structure

• Solve log₃(x−1)=2 over the reals.Automatically checked response; reasoning still matters
• Explain why ln(x²)=2 ln x is valid only for x>0 as written. Give a real-domain identity valid for every nonzero real x and test x=−2.Written reasoning: review required
• Solve ln(x−1)+ln(x+1)=ln 5 over the reals, giving exact roots and explaining every rejection.Written reasoning: review required

Fit growth models and change time units — Functions in graphs, measurements and growth decisions

• Prove equal differences for L(t)=A+mt and equal ratios for E(t)=Cb^t over any interval length h, with C nonzero and b>0. Explain why unequal gaps require adjustment.Written reasoning: review required

Translate recursive and explicit sequences — Functions in graphs, measurements and growth decisions

• A fund begins with 200 dollars at index 0. Model adding 20 dollars after each month versus multiplying the balance by 1.1 after each month. Give both recursive and explicit models, and compare at month 3.Written reasoning: review required

Read rates from uneven tables and graph scales — Functions in graphs, measurements and growth decisions

• A tank table gives (time minutes, volume liters): (0,12),(3,21),(8,31). What is the average rate from 3 to 8 minutes?Automatically checked response; reasoning still matters

Fit growth models and change time units — Functions in graphs, measurements and growth decisions

• A quantity grows by 8 percent each month. Give its model from initial value 50, its exact yearly factor and approximate yearly percentage; explain why 96 percent is not exact.Written reasoning: review required

Translate recursive and explicit sequences — Functions in graphs, measurements and growth decisions

• A fund begins with 200 dollars at index 0. Model adding 20 dollars after each month versus multiplying the balance by 1.1 after each month. Give both recursive and explicit models, and compare at month 3.Written reasoning: review required

Fit growth models and change time units — Functions in graphs, measurements and growth decisions

• Infer the exponential rule for supplied Graph H in days from its exact landmarks (0,20),(3,40),(6,80). Interpret the parameters and predict day 9 under the same ideal model.Written reasoning: review required

Translate recursive and explicit sequences — Functions in graphs, measurements and growth decisions

• A fund begins with 200 dollars at index 0. Model adding 20 dollars after each month versus multiplying the balance by 1.1 after each month. Give both recursive and explicit models, and compare at month 3.Written reasoning: review required

Use exponent laws without losing the base conditions — Algebra II and precalculus: functions and structure

• An original synthetic culture model N(t)=Ab^t has N(1)=30 and N(3)=120, with A>0 and b>0. Find A and b, then predict N(4). State what two observations cannot establish about a real culture.Written reasoning: review required

Find delayed exponential crossovers — Functions in graphs, measurements and growth decisions

• A graph shows 100n² above 2^n for n=4,…,12. A report says polynomial growth is always faster. Repair the claim using n=14,15 and a distinction between evidence and proof.Written reasoning: review required
• Prove that 3^n remains above 20n² for every integer n≥6 after the first crossing.Written reasoning: review required
• For fixed b>1, C>0 and positive integer k, outline why b^n eventually exceeds Cn^k. Why does this fail for b=1?Written reasoning: review required

Solve target times and justify rounding — Functions in graphs, measurements and growth decisions

• For Q(t)=2·10^(0.3t), solve Q(t)=9 and find the first integer time at or above 9. Supply neighboring checks.Written reasoning: review required
• A concentration is C(t)=100e^(−0.2t) for t≥0 hours. Solve C(t)≤30 and find the first whole-hour report satisfying it.Written reasoning: review required
• A model has Q(t)=40·2^(t/3), t in days. What is the first whole-number day n≥0 with Q(n)≥150?Automatically checked response; reasoning still matters

Fit growth models and change time units — Functions in graphs, measurements and growth decisions

• Infer the exponential rule for supplied Graph H in days from its exact landmarks (0,20),(3,40),(6,80). Interpret the parameters and predict day 9 under the same ideal model.Written reasoning: review required
• Rewrite P(t)=20·2^(t/3), t in days, using h in hours and w in weeks. Verify the same prediction at six days.Written reasoning: review required

Translate recursive and explicit sequences — Functions in graphs, measurements and growth decisions

• A fund begins with 200 dollars at index 0. Model adding 20 dollars after each month versus multiplying the balance by 1.1 after each month. Give both recursive and explicit models, and compare at month 3.Written reasoning: review required

Measure rotation and derive exact values — Trigonometry, polar geometry and precalculus representations

• An angle subtends a 15 cm arc on a circle of radius 6 cm. Give the angle in radians.Automatically checked response; reasoning still matters
• Write 225 degrees as kπ radians. Enter k exactly.Automatically checked response; reasoning still matters

Measure rotation and derive exact values — Trigonometry, polar geometry and precalculus representations

• Which unit-circle point corresponds to 2π/3?Automatically checked response; reasoning still matters
• A learner says −π/3 equals 5π/3. Repair the statement and explain why their sines and cosines agree.Written reasoning: review required

Measure rotation and derive exact values — Trigonometry, polar geometry and precalculus representations

• Derive sin(π/4) and cos(π/4) without a decimal calculator. Explain the signs for 5π/4.Written reasoning: review required
• Find sin(7π/6) exactly.Automatically checked response; reasoning still matters

Prove symmetry by moving a unit-circle point — School trigonometry: proofs, inverse branches and periodic decisions

• Start with an arbitrary point (c,s)=(cos x,sin x). Derive coordinates and all three trig values at π+x and 2π−x. State the tangent condition and explain why the argument is not restricted to the first quadrant.Written reasoning: review required

Prove symmetry by moving a unit-circle point — School trigonometry: proofs, inverse branches and periodic decisions

• Prove cosine is even and sine is odd on R. Prove the corresponding tangent statement on its domain, and explain why domain symmetry matters to a parity claim.Written reasoning: review required
• Prove 2π is a period of sine and cosine and π is a period of tangent. Then use suitable test angles to disprove π as a period of cosine and π/2 as a period of tangent. Distinguish a valid period from a proof of the least positive period.Written reasoning: review required

Fit a periodic model and test its assumptions — Trigonometry, polar geometry and precalculus representations

• A fictional wheel has radius5 m and centre6 m above ground, completes a revolution every20 seconds, and begins with a rider at the bottom. Write a height model and find heights at t=0,5,10.Written reasoning: review required
• Measurements at t=0,2,4 give outputs3,7,3. Does this prove a sinusoid with period4? Give a fitting cosine and explain what additional evidence is needed.Written reasoning: review required

Justify a principal inverse before using it — School trigonometry: proofs, inverse branches and periodic decisions

• Use the unit circle to justify why sine on [−π/2,π/2] and cosine on [0,π] are one-to-one and cover [−1,1]. Explain why using the full circle fails. Do not merely list memorized ranges.Written reasoning: review required
• Justify the tangent principal interval (−π/2,π/2) and range R by intersecting rays with the line X=1. Explain the missing endpoints and why secant is not the inverse tangent.Written reasoning: review required

Find every threshold time in a stated interval — School trigonometry: proofs, inverse branches and periodic decisions

• For a synthetic temperature model T(t)=18+5cos(πt/6), with t in hours, calculate every time T=20 on [0,15], to the nearest hundredth of an hour. State calculator mode, exact expressions and how you know the list is complete.Written reasoning: review required
• For the same temperature model, give every subinterval of [0,15] on which T≥20. Explain how endpoints and the initial value affect the answer rather than treating all crossings as rising.Written reasoning: review required
• A synthetic signal is S(t)=4−3sin(π(t+1)/5). Find all t in [0,12] with S(t)=5, reporting to 0.01 time units. Show how the negative amplitude and phase affect the equation; do not change the sine target sign without explanation.Written reasoning: review required

Prove identities on their shared domains — Trigonometry, polar geometry and precalculus representations

• If sin θ=5/13, what is cos²θ?Automatically checked response; reasoning still matters
• Given sin θ=3/5 and θ in quadrant II, what is cos θ?Automatically checked response; reasoning still matters
• Prove (1−cos²x)/sin x=sin x on its original domain and state that domain. Does cancellation make the original defined at x=0?Written reasoning: review required

Build addition formulas from rotated coordinates — School trigonometry: proofs, inverse branches and periodic decisions

• Without starting from a supplied addition identity, rotate the point (cos b,sin b) by a. Use the rotated horizontal and vertical unit directions to derive both sine and cosine addition formulas, explaining why the resulting angle is a+b.Written reasoning: review required
• Derive sine and cosine subtraction formulas from your rotation formulas and the reflection identities. Explain the different signs rather than memorizing a pattern.Written reasoning: review required
• Derive tan(a+b) and tan(a−b) as rational expressions in tan a and tan b. State all conditions for the expressions. Give one case where the trig value exists but the rational expression in tan a, tan b does not.Written reasoning: review required