LEARN · EXPLAIN · REVISE

Explain why an exponential model with base one is excluded from the usual increasing/decreasing classification.

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

高一必修第一册(A版).pdf · 4.2 · PDF 118 / printed page 111

Revisit first: Exponents

TOPIC 01

Exponential functions

Build understanding of exponential functions through definitions, contrasting cases and justified applications.

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

01 / Transfer#Worked example

f(x)=1^x

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Identify the base and whether the exponential graph grows or decays.

Hint 2

Check two different inputs.

Worked solution

It is the constant function 1.

Checks and common pitfalls: It is not strictly monotone and has no inverse on the full real line.

Reasoning checklist · self / teacher assessment

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

Teacher preparation and assessment

Interpret and use exponential functions with domain checks.
Base: A real exponential base is positive and different from one.
$y=a^x$
Growth and decay: Bases above one increase; bases between zero and one decrease.
Close with: conditions → representation → reasoning → check.

Expected reasoning: A real exponential base is positive and different from one.
Expected reasoning: Bases above one increase; bases between zero and one decrease.
Expected correction: An exponential never reaches its zero asymptote.