LEARN · EXPLAIN · REVISE

Can f(x)=x² on the real line have an inverse function? Give a counterexample.

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

2027 JM02 考試大綱 · 1. Functions · PDF 2 / printed page 2

TOPIC 01

Inverse functions and domain restrictions

Build understanding of inverse functions and domain restrictions through definitions, contrasting cases and justified applications.

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

01 / Standard#Your turn

f:\mathbb R\to[0,\infty);\quad f(x)=x^2

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Specify a one-to-one domain, solve for the input, then exchange the variables.

Hint 2

Find distinct inputs with the same output.

Worked solution

No: f(1)=f(−1)=1.

Checks and common pitfalls: Surjectivity onto the stated range alone does not give injectivity.

Reasoning checklist · self / teacher assessment

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

Teacher preparation and assessment

What must be true before using the main rule for inverse functions and domain restrictions?
Which representation makes this task easier, and why?
Change one assumption. Does the conclusion survive?

Choose an equivalent method and explain exclusions before calculating.
Undoing a rule: An inverse exchanges the original domain and range and requires a one-to-one rule.
$f^{-1}(f(x))=x$
Choosing a branch: Restricting a quadratic to either side of its vertex creates an invertible rule.
Close with: conditions → representation → reasoning → check.

Expected reasoning: An inverse exchanges the original domain and range and requires a one-to-one rule.
Expected reasoning: Restricting a quadratic to either side of its vertex creates an invertible rule.
Expected correction: A candidate produced by algebra may violate the original domain or sign condition.