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Functions: concept and properties: mixed assessment

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

TOPIC 01

Functions: concept and properties: 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 / Foundation#Your turn

Evaluate the function at the stated input.

f(x)=x^2-7x+1;\quad f(8)

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.
$f(8)=(8)^2-7(8)+1=9$
Function evaluation is different from solving f(x)=0.

The requested value is 9.

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

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

02 / Foundation#Your turn

Classify parity on the real line.

f(x)=x^3+7x

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the definition of the property and keep the domain in view.

Hint 2

Replace every x by −x.

Worked solution

Use the definition of the property and keep the domain in view.
Calculate or simplify this relation.
$f(-x)=-x^3-7x=-f(x)$
The domain is symmetric about zero.

Odd.

Checks and common pitfalls: The domain is symmetric about zero.

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 / Foundation#Your turn

Evaluate the real cube root.

(-343)^{1/3}

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 exponent, its real domain and how it changes order.

Hint 2

An odd root preserves the sign.

Worked solution

Identify the exponent, its real domain and how it changes order.
Calculate or simplify this relation.
$(-7)^3=-343$
Unlike an even root, a real odd root allows negative inputs.

The requested value is -7.

Checks and common pitfalls: Unlike an even root, a real odd root allows negative inputs.

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

04 / Foundation#Your turn

A service costs C=7+4n for n whole items. A bill is 27. Find n.

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Define the variable and its units, form a model, and check feasible inputs.

Hint 2

Subtract the fixed charge before dividing.

Worked solution

Define the variable and its units, form a model, and check feasible inputs.
Calculate or simplify this relation.
$4n=(27)-7=20\Rightarrow n=5$
The answer must be a nonnegative integer.

The requested value is 5.

Checks and common pitfalls: The answer must be a nonnegative integer.

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

05 / Foundation#Your turn

Evaluate the piecewise rule at the breakpoint.

f(x)=\begin{cases}x+1&x<8\\2x&x\ge8\end{cases};\quad f(8)

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.
$f(8)=2(8)=16$
The boundary belongs to exactly the second branch.

The requested value is 16.

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

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

06 / Foundation#Your turn

How does adding a constant affect monotonicity? Prove your answer.

g(x)=f(x)+8

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the definition of the property and keep the domain in view.

Hint 2

Compare two outputs by subtraction.

Worked solution

Use the definition of the property and keep the domain in view.
Calculate or simplify this relation.
$g(x_2)-g(x_1)=f(x_2)-f(x_1)$
The constant cancels from every output difference.

It preserves increasing or decreasing behaviour.

Checks and common pitfalls: The constant cancels from every output difference.

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.

07 / Standard#Your turn

Find m so that this is a power function with coefficient one.

f(x)=(m-8)x^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 exponent, its real domain and how it changes order.

Hint 2

The standard form is x raised to a fixed real power.

Worked solution

Identify the exponent, its real domain and how it changes order.
Calculate or simplify this relation.
$m-8=1\Rightarrow m=9$
A scalar multiple is not the stated coefficient-one power-function form.

The requested value is 9.

Checks and common pitfalls: A scalar multiple is not the stated coefficient-one power-function form.

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

08 / Standard#Your turn

Solve both parts and justify the conditions used. Part A: A learner fits a straight line through two observations and claims the model is valid forever. Evaluate the claim. Part B: Find the greatest value on the stated closed interval.

\begin{gathered}\text{B: }f(x)=(x-10)^2;\quad x\in[9,13]\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: Define the variable and its units, form a model, and check feasible inputs.

Hint 2

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

Worked solution

Part A reasoning
Define the variable and its units, form a model, and check feasible inputs.
Check further observations and the mechanism generating the data; state a justified range of use.
A model is conditional on assumptions and evidence.
Part B reasoning
Identify the input domain, rule and requested output before substituting.
Calculate or simplify this relation.
$f(9)=1,\quad f(10)=0,\quad f(13)=9$
The greatest distance from the vertex determines the maximum here.

A: Two observations determine a line but do not validate unlimited extrapolation. B: The requested value is 9.

Checks and common pitfalls: A model is conditional on assumptions and evidence. The greatest distance from the vertex determines the maximum here.

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.

09 / Standard#Your turn

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

f(x+9)=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.
$t=x+9;\quad f(t)=2(t-9)+3$
Renaming the dummy variable does not change the function.

f(x)=2x−15.

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.

10 / Standard#Your turn

State the monotonicity on the real line and justify it.

f(x)=-9x+2

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the definition of the property and keep the domain in view.

Hint 2

The slope is negative.

Worked solution

Use the definition of the property and keep the domain in view.
Calculate or simplify this relation.
$x_1<x_2\Rightarrow f(x_2)-f(x_1)=-9(x_2-x_1)<0$
The definition compares the outputs at ordered inputs.

Strictly decreasing.

Checks and common pitfalls: The definition compares the outputs at ordered inputs.

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

Solve both parts and justify the conditions used. Part A: Give the real domain and range. Part B: Evaluate the function at the stated input.

\begin{gathered}\text{A: }f(x)=x^{-2}\\\text{B: }f(x)=x^2-7x+1;\quad f(8)\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: Identify the exponent, its real domain and how it changes order.

Hint 2

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

Worked solution

Part A reasoning
Identify the exponent, its real domain and how it changes order.
Calculate or simplify this relation.
$x^{-2}=1/x^2>0\quad(x\ne0)$
Calculate or simplify this relation.
$y>0\Rightarrow x=1/\sqrt y\ne0,\quad f(x)=y$
Every positive output is attained by the displayed nonzero input; zero is excluded as both input and output.
Part B reasoning
Identify the input domain, rule and requested output before substituting.
Calculate or simplify this relation.
$f(8)=(8)^2-7(8)+1=9$
Function evaluation is different from solving f(x)=0.

A: Domain: x≠0; range: y>0. B: The requested value is 9.

Checks and common pitfalls: Every positive output is attained by the displayed nonzero input; zero is excluded as both input and output. Function evaluation is different from solving f(x)=0.

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.

12 / Standard#Your turn

A journey covers equal distances at speeds 18 and 54. Find its average speed.

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Define the variable and its units, form a model, and check feasible inputs.

Hint 2

Average speed is total distance divided by total time.

Worked solution

Define the variable and its units, form a model, and check feasible inputs.
Calculate or simplify this relation.
$v=\frac{2d}{d/18+d/54}=27$
Equal distances do not imply equal travel times.

The requested value is 27.

Checks and common pitfalls: Equal distances do not imply equal travel times.

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-10)^2;\quad x\in[9,13]

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.
$f(9)=1,\quad f(10)=0,\quad f(13)=9$
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.

14 / Transfer#Your turn

Find the minimum on the stated interval.

f(x)=(x-10)^2+2;\quad x\in[11,13]

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the definition of the property and keep the domain in view.

Hint 2

The vertex lies outside this interval.

Worked solution

Use the definition of the property and keep the domain in view.
Calculate or simplify this relation.
$x-10\in[1,3];\quad f(x)\ge1^2+2=3$
The unrestricted vertex value cannot be used as the constrained minimum.

The requested value is 3.

Checks and common pitfalls: The unrestricted vertex value cannot be used as the constrained minimum.

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

15 / Transfer#Your turn

Classify the parity of this power function on the reals.

f(x)=x^{20}

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 exponent, its real domain and how it changes order.

Hint 2

An even number of negative factors gives a positive product.

Worked solution

Identify the exponent, its real domain and how it changes order.
Calculate or simplify this relation.
$f(-x)=(-x)^{20}=x^{20}=f(x)$
The exponent is an even integer and the domain is symmetric.

Even.

Checks and common pitfalls: The exponent is an even integer and the domain is symmetric.

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.

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

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.