LEARN · EXPLAIN · REVISE

Are these the same function with their natural real domains? Explain.

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

高一必修第一册(A版).pdf · 3.1 · PDF 67 / printed page 60

TOPIC 01

Function concept and representations

Build understanding of function concept and representations 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(x)=\frac{x^2-1}{x-1};\quad g(x)=x+1

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Identify the input domain, rule and requested output before substituting.

Hint 2

Simplification does not restore a forbidden input.

Worked solution

No: f excludes x=1, whereas g does not.

Checks and common pitfalls: A function includes its domain as well as its rule.

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 function concept and representations?
Which representation makes this task easier, and why?
Change one assumption. Does the conclusion survive?

Interpret and use function concept and representations with domain checks.
Function: Each allowed input has exactly one output.
Domain and range: The domain is specified or restricted by the rule; the range is the set of attained outputs.
Close with: conditions → representation → reasoning → check.

Expected reasoning: Each allowed input has exactly one output.
Expected reasoning: The domain is specified or restricted by the rule; the range is the set of attained outputs.
Expected correction: Equal formulas on some inputs need not define equal functions.