Week 3 — What Goes In, What Comes Out? A Short Function Report

Assignment 3 of the term — evaluate a function at numbers and at an expression, read a domain and a range in a real setting, compose two functions in the right order, and explain a composition to someone new to algebra.

This adaptive assignment uses a coaching dialogue to help you turn a first attempt into a clear, checkable report. You remain the author. The coach asks questions, checks the reasoning you show, and waits for you to revise before it produces the required report. In Week 1 you evaluated expressions with the order of operations and learned that (−2)² = 4 while −2² = −4; this week that same evaluation gets a name, function notation, and a new question: which inputs a rule accepts and in what order two rules run. This week's lecture tutorial, quiz, discussion, and Model/Build Studio practise the same ideas from other angles; this assignment is where you put them together.

The task

This week's two named functions are f(x) = 3x² − 2x + 5, whose outputs at the inputs −2, −1, 0, 1, 2, 3 are 21, 10, 5, 6, 13, 26, and g(x) = 3x + 4, a bowl shop's price in dollars for x toppings (4 dollars for the bowl plus 3 dollars per topping). The shop sells 0 to 5 toppings, so its practical domain is the whole numbers from 0 to 5. The launched object's height in feet is h(t) = −16t² + 64t + 80, t seconds after launch; its heights at t = 0, 1, 2, 3, 4, 5 are 80, 128, 144, 128, 80, 0, and it lands at t = 5.

Work these four problems, showing your arithmetic:

1. Let f(x) = 3x² − 2x + 5. Find f(−2).
2. A bowl shop charges g(x) = 3x + 4 dollars for x toppings. A friend orders 2 more toppings than you, and you order a = 3 toppings. Write g(a + 2) in simplest form, then find the friend's price when a = 3.
3. The launched object's height in feet is h(t) = −16t² + 64t + 80, t seconds after launch, and it lands at t = 5. Find h(2), the height at the top of the flight.
4. Let f(x) = 3x² − 2x + 5 and g(x) = 3x + 4. Find (g ∘ f)(−2).

Beside each answer, add one brief interpretation: what the value means, which rule or step produced it, and one use or limit of it. For problem 2, say for which values of a the friend's order stays inside the shop's practical domain. For problem 3, state the practical domain and the range of h and say which one comes from the times and which from the heights. For problem 4, also find (f ∘ g)(−2), compare it with your answer, and explain which function is the inner function in each.

Explain it to a non-expert

A relative who runs the register at the bowl shop has never studied algebra. They ask, "When a student uses the 2-dollar discount, does it matter whether I take the 2 off before or after I price the toppings?" Write a 3–5 sentence message that answers in everyday words, gives a reason tied to the shop's prices (you may use a 3-topping order), and names one limit — for example, that taking 2 off the topping count would only make sense for an order of at least 2 toppings, or that the answer depends on the shop's actual rule. Explain any technical term, such as "input" or "composition", in everyday words. More than one good answer is possible.

For a different, ungraded example, imagine a bike-repair volunteer who wonders whether a shop should add a service fee before or after a member discount. Model answer: "If the shop adds the fee first and then takes the member discount off the whole bill, the member also gets a discount on the fee. If it takes the discount off first and adds the fee after, the fee stays full price. So the order changes what the member pays. The difference is small for a cheap repair, and the shop's posted policy decides which one is fair to promise."

Use that pattern for the bowl shop, with your own explanation and evidence.

Part 1 — Student instructions

1. Draft all four answers, the four brief interpretations, and your message to the relative before opening a chatbot.
2. Open any course-approved chatbot. Copy the entire Coach Prompt box, including its first and last marker lines, and paste it as one message. The coach will ask for your draft and then work one question at a time.
3. Answer in your own words. Show calculations when asked. Use the coach's feedback to revise at least one calculation or explanation, even if your first attempt was correct.
4. Continue until the prompt's exit condition is met.
5. Save the chat's share link and copy the final report exactly as the coach gives it.

The chatbot is a coach, not a ghostwriter. It must draw out your reasoning rather than write the report for you.

Coach Prompt

COPY EVERYTHING BELOW THIS LINE
ROLE: You are a careful learning coach. Draw out and challenge MY thinking about function notation, domain and range, and composition. Do not write my assignment for me and do not invent work I did not show.

ASSIGNMENT: Work through these four problems in order, one at a time. Do not invent, replace, or reorder a problem. The data: f(x) = 3x^2 - 2x + 5, whose outputs at -2, -1, 0, 1, 2, 3 are 21, 10, 5, 6, 13, 26; g(x) = 3x + 4, a bowl shop's price in dollars for x toppings, with a practical domain of the whole numbers 0 to 5; and the launched object h(t) = -16t^2 + 64t + 80, with heights 80, 128, 144, 128, 80, 0 feet at t = 0, 1, 2, 3, 4, 5 seconds, landing at t = 5. In this box, ^2 means squared and (g o f)(x) means g(f(x)).
PROBLEM P1: Let f(x) = 3x² − 2x + 5. Find f(−2).
PROBLEM P2: A bowl shop charges g(x) = 3x + 4 dollars for x toppings. A friend orders 2 more toppings than you, and you order a = 3 toppings. Write g(a + 2) in simplest form, then find the friend's price when a = 3.
PROBLEM P3: The launched object's height in feet is h(t) = −16t² + 64t + 80, t seconds after launch, and it lands at t = 5. Find h(2), the height at the top of the flight.
PROBLEM P4: Let f(x) = 3x² − 2x + 5 and g(x) = 3x + 4. Find (g ∘ f)(−2).
After the four answers, I must briefly interpret each one: what the value means, which rule or step produced it, and one use or limit of it; for P2, say for which values of a the friend's order stays in the practical domain; for P3, state the practical domain and the range of h; for P4, also find (f o g)(-2), compare it with my answer, and name the inner function in each. I must then write a 3-5 sentence message to a relative who runs the bowl-shop register and is new to algebra, explaining whether the order of a 2-dollar discount and the topping price matters, giving a reason tied to the shop's prices, and naming one limit. I must explain technical terms in everyday words.

SCORING CRITERIA: Score only evidence I supply: correct function computations (40 points), function reasoning and interpretation (32 points), clarity for a non-expert reader (16 points), and showing work so it can be checked (12 points). For each of P1-P4, give 2, 1, or 0 evidence units per criterion; one unit is worth 5 computation points, 4 reasoning points, 2 clarity points, or 1.5 showing-work points. Sum all four problems within each criterion. Count my final message only with P4, not again with P1-P3.
COMPUTATION UNITS: A correct value = 2; the right values and operations with only an arithmetic slip = 1; a missing result or the wrong values or operations = 0. Do not penalize an alternative valid method or equivalent correct notation.
REASONING UNITS: For P1-P3 separately, an accurate account of what the value means and which rule produced it plus a use or limit tied to the setting = 2; exactly one of the two = 1; neither = 0. For P4, BOTH an accurate order check that compares (g o f)(-2) with (f o g)(-2) AND a final message with a reason tied to the bowl-shop prices and a named limit = 2. If full P4 evidence is absent but at least one accurate substantive part appears, give 1. No accurate reasoning = 0; naming a rule alone does not earn a unit. Squaring -2 as -4, failing to distribute, reading a range as a domain, or treating a composition as a product does not count as accurate. An arithmetic slip alone is not a reasoning error.
CLARITY UNITS: For each problem, labeled evidence understandable to someone new to algebra, with technical terms explained = 2; intelligible evidence that needs rereading because of ambiguity or jargon = 1; absent or unintelligible evidence = 0. Include the final message with P4 if submitted. Missing reasoning alone does not lower clarity for otherwise clear submitted work.
SHOWING-WORK UNITS: For each problem, checkable values and steps = 2; a relevant but incomplete value or step = 1; a result alone or no method = 0. A visible method stays checkable despite an arithmetic slip.

Do NOT invent grading rules.

DIALOGUE RULES: Ask exactly one question per message and wait for my answer. Start by asking for my complete first attempt. Build each next question from my exact words. Probe an unsupported claim and ask me to examine its evidence. Keep the conversation focused on questions rather than completed work.

ENGAGEMENT GUARDS: A greeting, copied prompt, yes/no reply, or one-word answer is not a substantive exchange. Do not flatter or accept agreement as evidence. Praise only a concrete move I made. If evidence is missing, say what is missing and ask for it. Require a real revision plus a short reflection on why the revision is better. Do not issue the report early.

EXIT CONDITION: Continue until at least five substantive exchanges have occurred AND I have supplied all four answers and their brief interpretations AND written my message to the relative AND revised at least one calculation or explanation AND reflected on the revision. Stop only after whichever requirement happens last.

REPORT: After the exit condition is satisfied, output exactly these labelled lines in this order. Replace X with the exact earned score; it may end in .5 and must not be rounded to a whole number. Base every line only on evidence in this chat; write "Not supplied" for missing evidence and never raise the score for polished chatbot prose.
STUDENT'S SCORE: X/100
EVIDENCE FOR THE SCORE: [specific evidence from my calculations and reasoning]
REVISION MADE: [what I changed and why]
STRENGTH: [one earned strength]
NEXT STEP: [one concrete improvement]
Then tell me to submit this report, my revised function report, and the share link.
COPY EVERYTHING ABOVE THIS LINE

Deliverables

Submit these three items in the assignment entry:

• your revised function report with all four answers, their brief interpretations, and your message to the relative — written, or recorded as audio or video with a transcript, or as a labelled sketch of the launched object's flight or a small input-output table with a short note; every form is graded by the same scoring criteria;
• the coach's five-line report; and
• the conversation share link.

Submit in any accessible file type or as a link; a recorded response may be submitted as audio or video with a transcript, or as text.

Equivalent route (no AI required)

This route assesses the same Week 3 skills — evaluating a function at numbers and at an expression, reading a practical domain and a range, and composing two functions in the right order — for comparable time and effort, about 60 to 90 minutes, the same as the coached route. Complete the same four problems, the four brief interpretations, and the message to the relative. Then audit your own first draft against the attached scoring criteria, criterion by criterion, and write a second draft. Add a 150–200 word revision memo that names one calculation or explanation you changed, the evidence that prompted the change, and one question you still have. Submit the revised function report, your completed criteria checklist, and the revision memo in the assignment entry; together they replace the share link and the coach's report. This route uses the same scoring criteria and the same points, with no penalty for choosing it.

How you'll be graded

The full analytic rubric is attached to this assignment in Canvas.

Academic integrity and AI use

The calculations, reasoning, revision, and final report must be your own. Do not ask the chatbot to create the assignment or add evidence you did not produce. Submit an honest share link from your own dialogue. The equivalent route keeps the same workload and evidence and requires no AI account.

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