LEARN · EXPLAIN · REVISE

Function concept and representations

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

Revisit first: Quadratic functions, equations and inequalities

TOPIC 01

Function concept and representations

Build understanding of function concept and representations through definitions, contrasting cases and justified applications.

What you will be able to explain

Interpret and use function concept and representations with domain checks.
Connect representations and justify the steps, including boundary cases.
Explain a solution and apply the idea to a changed situation.

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.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Before calculating, predict how the conclusion changes when one defining condition in function concept and representations changes. Record a reason.

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

Quadratic graph: horizontal shift 0, vertical shift 0. Input is replaced by x−0.

Explain: Compare two admissible cases and one boundary or invalid case. Explain the observed difference using the stated definition.

Transfer: Construct a new example and a tempting incorrect solution. Repair the solution by naming the missing condition.

Try it. Leave your reasoning visible.

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

01 / Foundation#Worked example

Evaluate the function at the stated input.

f(x)=x^2-2x+1;\quad f(3)

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

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

Hint 2

Substitute the whole input into every occurrence of x.

Worked solution

Identify the input domain, rule and requested output before substituting.
Calculate or simplify this relation.
$f(3)=(3)^2-2(3)+1=4$
Function evaluation is different from solving f(x)=0.

The requested value is 4.

Checks and common pitfalls: Function evaluation is different from solving f(x)=0.

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

02 / Standard#Worked example

Find the real domain.

f(x)=\frac{\sqrt{x-2}}{x-4}

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

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

Hint 2

Combine the square-root and denominator restrictions.

Worked solution

Identify the input domain, rule and requested output before substituting.
Calculate or simplify this relation.
$x-2\ge0,\quad x-4\ne0$
Zero is allowed under a square root but not in a denominator.

[2,∞) excluding 4.

Checks and common pitfalls: Zero is allowed under a square root but not in a denominator.

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.

03 / Transfer#Worked example

Recover f(x) from the shifted-input identity.

f(x+2)=2x+3

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

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

Hint 2

Introduce a new variable for the complete input.

Worked solution

Identify the input domain, rule and requested output before substituting.
Calculate or simplify this relation.
$t=x+2;\quad f(t)=2(t-2)+3$
Renaming the dummy variable does not change the function.

f(x)=2x−1.

Checks and common pitfalls: Renaming the dummy variable does not change the function.

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.

04 / Foundation#Your turn

Evaluate the function at the stated input.

f(x)=x^2-3x+1;\quad f(4)

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

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

Hint 2

Substitute the whole input into every occurrence of x.

Worked solution

Identify the input domain, rule and requested output before substituting.
Calculate or simplify this relation.
$f(4)=(4)^2-3(4)+1=5$
Function evaluation is different from solving f(x)=0.

The requested value is 5.

Checks and common pitfalls: Function evaluation is different from solving f(x)=0.

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

05 / Foundation#Your turn

Find the real domain.

f(x)=\frac{\sqrt{x-3}}{x-5}

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

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

Hint 2

Combine the square-root and denominator restrictions.

Worked solution

Identify the input domain, rule and requested output before substituting.
Calculate or simplify this relation.
$x-3\ge0,\quad x-5\ne0$
Zero is allowed under a square root but not in a denominator.

[3,∞) excluding 5.

Checks and common pitfalls: Zero is allowed under a square root but not in a denominator.

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.

06 / Foundation#Your turn

Find the greatest value on the stated closed interval.

f(x)=(x-3)^2;\quad x\in[2,6]

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

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

Hint 2

Compare the vertex and both endpoints.

Worked solution

Identify the input domain, rule and requested output before substituting.
Calculate or simplify this relation.
$f(2)=1,\quad f(3)=0,\quad f(6)=9$
The greatest distance from the vertex determines the maximum here.

The requested value is 9.

Checks and common pitfalls: The greatest distance from the vertex determines the maximum here.

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

07 / Foundation#Your turn

Evaluate the piecewise rule at the breakpoint.

f(x)=\begin{cases}x+1&x<3\\2x&x\ge3\end{cases};\quad f(3)

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

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

Hint 2

Read which branch includes equality.

Worked solution

Identify the input domain, rule and requested output before substituting.
Calculate or simplify this relation.
$f(3)=2(3)=6$
The boundary belongs to exactly the second branch.

The requested value is 6.

Checks and common pitfalls: The boundary belongs to exactly the second branch.

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

08 / Standard#Your turn

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

f(x)=\frac{x^2-1}{x-1};\quad g(x)=x+1

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

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

Identify the input domain, rule and requested output before substituting.
Calculate or simplify this relation.
$f(x)=x+1\quad(x\ne1)$
A function includes its domain as well as its rule.

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

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.

09 / Standard#Your turn

Does the relation define y as a function of x? Explain.

\{(1,3),(1,4),(2,5)\}

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

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

Hint 2

Check uniqueness of the output for each input.

Worked solution

Identify the input domain, rule and requested output before substituting.
Calculate or simplify this relation.
$1\mapsto3,\quad1\mapsto4$
Different inputs may share an output; one input may not have two outputs.

No: input 1 has two different outputs.

Checks and common pitfalls: Different inputs may share an output; one input may not have two outputs.

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.

10 / Standard#Your turn

Recover f(x) from the shifted-input identity.

f(x+3)=2x+3

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

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

Hint 2

Introduce a new variable for the complete input.

Worked solution

Identify the input domain, rule and requested output before substituting.
Calculate or simplify this relation.
$t=x+3;\quad f(t)=2(t-3)+3$
Renaming the dummy variable does not change the function.

f(x)=2x−3.

Checks and common pitfalls: Renaming the dummy variable does not change the function.

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.

11 / Standard#Your turn

Evaluate the function at the stated input.

f(x)=x^2-4x+1;\quad f(5)

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

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

Hint 2

Substitute the whole input into every occurrence of x.

Worked solution

Identify the input domain, rule and requested output before substituting.
Calculate or simplify this relation.
$f(5)=(5)^2-4(5)+1=6$
Function evaluation is different from solving f(x)=0.

The requested value is 6.

Checks and common pitfalls: Function evaluation is different from solving f(x)=0.

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

12 / Transfer#Your turn

Find the real domain.

f(x)=\frac{\sqrt{x-4}}{x-6}

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

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

Hint 2

Combine the square-root and denominator restrictions.

Worked solution

Identify the input domain, rule and requested output before substituting.
Calculate or simplify this relation.
$x-4\ge0,\quad x-6\ne0$
Zero is allowed under a square root but not in a denominator.

[4,∞) excluding 6.

Checks and common pitfalls: Zero is allowed under a square root but not in a denominator.

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.

13 / Transfer#Your turn

Find the greatest value on the stated closed interval.

f(x)=(x-4)^2;\quad x\in[3,7]

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

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

Hint 2

Compare the vertex and both endpoints.

Worked solution

Identify the input domain, rule and requested output before substituting.
Calculate or simplify this relation.
$f(3)=1,\quad f(4)=0,\quad f(7)=9$
The greatest distance from the vertex determines the maximum here.

The requested value is 9.

Checks and common pitfalls: The greatest distance from the vertex determines the maximum here.

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

End-of-lesson check and correction

Review your latest checked answers and explanations. A draft change requires a fresh check. Written work needs your self-assessment or a teacher’s review.

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Choose a foundation skill to revisit ↗

Teacher preparation and assessment

Question sequence

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?

Board plan

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.

Anticipated thinking

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.

Assessment checklist

1: identify the givens and required quantity.
1: choose a valid definition, representation or method.
1: present connected, correct reasoning.
1: check conditions and explain the result.