Kenneth ChengMathematical modeling
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Find the inverse function and its domain.

Read the idea, work independently, then explain what changed.

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TOPIC 01

Exponential and logarithmic functions: mixed assessment

A mixed assessment: identify the method, justify it and revise your reasoning.

What you will be able to explain

  • Connect the chapter skills without relying on the order of the exercises.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Transfer#Your turn

Find the inverse function and its domain.

y=log⁡9xy=\log_9x
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use logarithmic monotonicity after enforcing a positive argument.
Hint 2
Swap input and output after solving for the original input.
Worked solution

  1. Use logarithmic monotonicity after enforcing a positive argument.

  2. Calculate or simplify this relation.

    x=9y⇒f−1(x)=9xx=9^y\Rightarrow f^{-1}(x)=9^x

  3. The original logarithm range becomes the inverse domain.

Inverse: y=9^x for all real x.

Checks and common pitfalls: The original logarithm range becomes the inverse domain.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

Focus on one question

Teacher preparation and assessment

Question sequence

  • Ask students to name the relevant condition before calculating.

Board plan

  • Compare valid methods and annotate their conditions.

Anticipated thinking

  • A correct final value may still hide a missing assumption.

Assessment checklist

  • Check the method, conditions, reasoning and interpretation separately.

No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.

Curriculum and source notes ↗