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Problem 46

The Observatory's Inside-Out Signal

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The Observatory's Inside-Out Signal

Complete to collectSunstone CabochonLevel 3 · 5 Glints

Build the signal forward

In the Observatory, one instrument turns a raw signal into a displayed echo. All functions take and return real numbers whenever they are used.

At time , the raw signal is . The instrument acts in this order:

  1. The delay gate uses the raw reading from 3 seconds earlier.
  2. The lift lens doubles its input and adds 5.
  3. The calibration lens applies a function . The function has an inverse .

The final displayed echo is .

Three Observatory stages transform one signal.

Forward equations

Write your equations and reasoning on separate paper.

  1. Let be the delay map. Define .

  2. Let be the raw signal after the delay gate. Write one equation using and , then write it again without .

  3. Let be the lift lens. Define .

  4. Let be the signal after the lift lens. Write one equation using and , then expand it using .

  5. Write in terms of and .

  6. Write one fully nested equation for using , , and .

  7. Write the same instrument as one composition equation using , , , and . State which function acts first.

Page 2 · Run the signal inside-out

Recover the raw signal

The forward instrument delays, lifts, and calibrates. Now use its displayed output to reconstruct what entered the instrument.

Reverse equations

Continue on separate paper.

  1. Find the inverse of the lift lens. Write a formula for , and verify it by simplifying .

  2. Use the future display to recover the current raw signal . Write:

    • one equation using and ;
    • the same equation with the formula for substituted;
    • one sentence explaining why must act before .

At a glance

About this problem

These tags show what the problem is about and the math you’ll use. Choose one to find similar problems.

Theme

Skills

Helpful first

Grade

Math area

Topics

Is this a good fit?

Before you choose this problem

A plain-language look at readiness, thinking, time, and the expected response.

Math readiness

Designed for Grade 10 · Grade 11 · Grade 12

Uses HS A2 math.

Thinking

Stretch challenge

A nontrivial plan using familiar mathematical ideas.

Time & response

About 11–28 minutes

Moderate amount of written work · Extended Explanation response.

How to support: The useful struggle is tracking which function acts first in the story and which inverse must act first during recovery. Ask the learner to narrate the stages aloud before changing notation.

How did it feel?

Pick the option that best matches this challenge. Recommendations use topic, math area, grade, story, and thinking challenge.

Keep exploring

The Manyfold challenge 14 of 22

Technical reference

Rating, skills & curriculum

Designed for Grade 10 · Grade 11 · Grade 12 · Uses HS A2 math · About 11–28 minutes

Grade 10 · Grade 11 · Grade 12 · Functions and change · Transformations, composition, and inverses

MJ HS:A2.6 · C3 · W2

Precise MJD coordinates, factor scores, skill weights, calibration evidence, and curriculum crosswalks. This reference is available to everyone and is intentionally last.

Content level

MJ HS:A2.6

High school

HS A2 · point 6

Likely range
MJ HS:A2.5–HS:A2.7
Confidence
84%

Challenge

C3 · Stretch

A nontrivial plan using familiar mathematical ideas.

  • Method selection2
  • Reasoning depth3
  • Novelty2
  • Constraint management3
  • Proof or justification2

Workload

W2 · Moderate

Several coordinated calculations, representations, or explanation steps.

Active time
11–28 minutes · typically 18
Response
Extended Explanation
Known-method steps
19
  • Mechanical execution1
  • Written output3
  • Representation production2
  • Bookkeeping2

Learning focus

Skills this problem practices

What a successful response demonstrates, with each skill’s share of the content rating.

  • Compose functions and interpret composition

    The learner can compose functions and interpret composition. This capability is distinct from mere exposure to or isolated use of function notation.

    • Independent mastery
    • 55% focus
    • MJ HS:A2.5 milestone

  • Find and verify inverse functions on appropriate domains

    The learner can find and verify inverse functions on appropriate domains. This capability is distinct from mere exposure to or isolated use of composition; one-to-one.

    • Independent mastery
    • 45% focus
    • MJ HS:A2.6 milestone

Readiness

Helpful prerequisites

Skills the rating assumes are already available to the learner.

  • Use function notation and determine domain and range in context

    The learner can use function notation and determine domain and range in context. This capability is distinct from mere exposure to or isolated use of Grade 8 functions.

    • Fluent readiness
    • Required
    • MJ HS:A1.6 milestone

  • Understand a function as assigning exactly one output to each input

    The learner can understand a function as assigning exactly one output to each input. This capability is distinct from mere exposure to or isolated use of variables; relations.

    • Fluent readiness
    • Required
    • MJ 8.5 milestone

  • Solve literal equations and formulas for a variable

    The learner can solve literal equations and formulas for a variable. This capability is distinct from mere exposure to or isolated use of rearrange formulas.

    • Independent readiness
    • Required
    • MJ HS:A1.4 milestone

External alignment

Curriculum and standards crosswalks

Approved evidence-backed locations that support discovery. They do not define the MathJewels rating.

Common Core State Standards for Mathematics2010 · 2 locations
Clusters2 locations · Representative: Build a function that models a relationship between two quantities