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Solve both parts and justify the conditions used. Part A: Explain why the graphs y=log₂x and y=2ˣ are reflections in y=x. Part B: 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 / Standard#Your turn

Solve both parts and justify the conditions used. Part A: Explain why the graphs y=log₂x and y=2ˣ are reflections in y=x. Part B: Find the inverse function and its domain.

\begin{gathered}\text{A: }y=\log_2x\iff x=2^y\\\text{B: }y=\log_9x\end{gathered}

Use the domain, units and sampling assumptions stated in the question.
Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

A: Use logarithmic monotonicity after enforcing a positive argument.

Hint 2

B: Use logarithmic monotonicity after enforcing a positive argument.

Worked solution

Part A reasoning
Use logarithmic monotonicity after enforcing a positive argument.
Calculate or simplify this relation.
$(a,b)\mapsto(b,a)$
The axes and domains are exchanged, not simply translated.
Part B reasoning
Use logarithmic monotonicity after enforcing a positive argument.
Calculate or simplify this relation.
$x=9^y\Rightarrow f^{-1}(x)=9^x$
The original logarithm range becomes the inverse domain.

A: The input-output coordinates are exchanged by inversion. B: Inverse: y=9^x for all real x.

Checks and common pitfalls: The axes and domains are exchanged, not simply translated. The original logarithm range becomes the inverse domain.

Reasoning checklist · self / teacher assessment

Solve part A with its stated restrictions.
Solve part B using an appropriate representation.
Give the reasoning and check conditions in both parts.

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

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.