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Week 3 · Assignment: Traditional alternative (no AI)

Week 3 — What Goes In, What Comes Out? A Short Function Report — Traditional alternative (no AI)

College Algebra Free sample edition
Traditional alternative (no AI), unpublished. In the course this is a separate assignment that ships unpublished beside the adaptive assignment: same objective, no AI. An instructor who prefers it publishes it and unpublishes or deletes the adaptive one, as the note at its top says.

Instructors: publish this instead of the adaptive version if you prefer, and unpublish or delete the adaptive one.

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 week you learned to recognize a function, to use function notation, to evaluate at numbers and at expressions, to find a domain and read a range, and to combine and compose two functions. 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 and a new question: which inputs a rule accepts and in what order two rules run. In this assignment you work four short problems, interpret each one, and explain a composition to someone who has never studied algebra. This week's lecture tutorial, quiz, discussion, and Model/Build Studio practise the same ideas from other angles.

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.

Show your arithmetic. A reader should be able to reproduce every number you report from the values above without guessing which values or steps you used.

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.

Deliverables

Submit a single function report containing:

  • the four answers, each with the arithmetic shown and a brief interpretation of what the value means, which rule or step produced it, and one use or limit of it;
  • your message to the relative, explaining whether the order matters with a reason tied to the shop's prices and naming one limit; and
  • a one-line restatement of the functions and inputs you used, so your results are reproducible.

Keep it to roughly one page. Clear and correct beats long.

Choose how you show it. Present your report as written text, or as a recorded audio or video explanation of about three minutes 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. 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.

How you'll be graded

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

Academic integrity and AI use

This version is designed to be completed without AI: do the evaluations, the reasoning, the interpretations, and the writing yourself, without a chatbot or other AI tool. You may use a calculator or a graphing calculator such as Desmos to check the arithmetic. Submitting AI-generated work as your own, or copying another student's numbers, is an integrity violation under the course policy.

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