PorismaSource review

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Source review · Functions

All source-review topics · Coverage and provenance · Student practice

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17 question records

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04 · Functions

Edition metadata →

04-01 · When can two functions compose?Question pp.1 / Solution pp.1, 2, 3

Question and explanations → · Complete native source article →

[2023/MI/Promo/PU2/I/8]. Printed/task labels inventoried: i, ii, iii, iv, v.

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2 page-item records.

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4 page-item records.

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5 page-item records. Native SVG generated; rendering is checked separately

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Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
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Endpoint checks alone are insufficient: the minimum of g occurs inside its domain.

Essential answer conditions and source qualifications

$0\le x\le4$ .

Exact private source page references

04 Functions sol.pdf · p.1
04 Functions sol.pdf · p.2
04 Functions sol.pdf · p.3
04 Functions.pdf · p.1

04-02 · Invert, then repeat periodicallyQuestion pp.1 / Solution pp.4

Question and explanations → · Complete native source article →

[2023/ACJC/Promo/5]. Printed/task labels inventoried: a.i, a.ii, b.i, b.ii.

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2 page-item records.

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1 page-item record.

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1 page-item record. Native SVG generated; rendering is checked separately

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Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

No separate condition recorded. Check the full question, native working and printed assumptions.

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04 Functions sol.pdf · p.4
04 Functions.pdf · p.1

04-03 · Restrict the domain to choose an inverseQuestion pp.1, 2 / Solution pp.5, 6, 7

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[2011/IJC/Prelim/I/9]. Printed/task labels inventoried: i, ii, iii, iv, v.

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3 page-item records.

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6 page-item records.

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3 page-item records. Native SVG generated; rendering is checked separately

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Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

The general claim that f and its inverse intersect only on y=x is false for decreasing functions. The working here includes the extra admissibility check.

Essential answer conditions and source qualifications

Equal coordinate-unit scales.

For the restricted function; general decreasing functions can have additional inverse-intersection two-cycles.

Exact private source page references

04 Functions sol.pdf · p.5
04 Functions sol.pdf · p.6
04 Functions sol.pdf · p.7
04 Functions.pdf · p.1
04 Functions.pdf · p.2
04 Functions.pdf · p.8

04-04 · A composite can reach an interior minimumQuestion pp.2 / Solution pp.8, 9

Question and explanations → · Complete native source article →

[modified 2011/HCI/Prelim/II/2]. Printed/task labels inventoried: a.i, a.ii, b.i, b.ii.

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2 page-item records.

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2 page-item records.

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4 page-item records.

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4 page-item records. Native SVG generated; rendering is checked separately

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Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

No separate condition recorded. Check the full question, native working and printed assumptions.

Exact private source page references

04 Functions sol.pdf · p.8
04 Functions sol.pdf · p.9
04 Functions.pdf · p.2

04-05 · Identity rules still have domainsQuestion pp.2 / Solution pp.10

Question and explanations → · Complete native source article →

[2011/DHS/Prelim/I/4]. Printed/task labels inventoried: a, b.i, b.ii, b.iii.

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2 page-item records.

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1 page-item record.

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3 page-item records.

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4 page-item records. Native SVG generated; rendering is checked separately

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Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

b(ii)

Domains x≥1, x≥0, x≥1 respectively.

Exact private source page references

04 Functions sol.pdf · p.10
04 Functions.pdf · p.2
04 Functions.pdf · p.8

04-06 · Three compositions return the inputQuestion pp.2 / Solution pp.11

Question and explanations → · Complete native source article →

[2014/JJC/Promo/8]. Printed/task labels inventoried: i, ii, iii, iv.

Question / subparts

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2 page-item records.

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Solution steps

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1 page-item record.

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1 page-item record.

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Tables

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0 page-item records.

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Diagrams

Source comparison recorded

1 page-item record. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

iii

$x\ne0,1$ .

$x>0$ .

Exact private source page references

04 Functions sol.pdf · p.11
04 Functions.pdf · p.2

04-07 · Absolute values and the correct inverse branchQuestion pp.3 / Solution pp.12, 13

Question and explanations → · Complete native source article →

[2011/AJC/Promo/8]. Printed/task labels inventoried: i, ii, iii.

Question / subparts

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2 page-item records.

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2 page-item records.

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Teacher notes

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3 page-item records.

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Tables

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0 page-item records.

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Diagrams

Source comparison recorded

2 page-item records. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

No separate condition recorded. Check the full question, native working and printed assumptions.

Exact private source page references

04 Functions sol.pdf · p.12
04 Functions sol.pdf · p.13
04 Functions.pdf · p.3

04-08 · Invert a special point separatelyQuestion pp.3 / Solution pp.13, 14

Question and explanations → · Complete native source article →

[2010/MJC/Prelim/I/10]. Printed/task labels inventoried: i, ii, iii.

Question / subparts

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2 page-item records.

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2 page-item records.

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Teacher notes

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3 page-item records.

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Tables

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0 page-item records.

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Diagrams

Source comparison recorded

2 page-item records. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

Maclaurin expansion near x=0.

iii — inverse

Continuous inverse branch x>−ln2; isolated inverse input x=−1.

Exact private source page references

04 Functions sol.pdf · p.13
04 Functions sol.pdf · p.14
04 Functions.pdf · p.3

04-09 · The decreasing sine branchQuestion pp.3 / Solution pp.15

Question and explanations → · Complete native source article →

[2016/NYJC/Prelim/I/10a]. Printed/task labels inventoried: i, ii.

Question / subparts

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2 page-item records.

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Solution steps

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1 page-item record.

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Teacher notes

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1 page-item record.

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Tables

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0 page-item records.

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Diagrams

Source comparison recorded

1 page-item record. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed-with-source-correction. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Source discrepancy: solution PDF p.15 prints arcsin(x/2)−π/3. This maps 2 to π/6 but maps −2 to −5π/6, outside the required domain. Porisma shows the corrected branch and retains the original source.

Essential answer conditions and source qualifications

ii — inverse

Correct decreasing branch; printed arcsin−π/3 rule is wrong.

Exact private source page references

04 Functions sol.pdf · p.15
04 Functions.pdf · p.3

04-10 · Equal identity functions need equal domainsQuestion pp.4 / Solution pp.16, 17

Question and explanations → · Complete native source article →

[2021/EJC/Promo/8]. Printed/task labels inventoried: i, ii, iii, iv, v.

Question / subparts

Source comparison recorded

2 page-item records.

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Solution steps

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3 page-item records.

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Teacher notes

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3 page-item records.

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Tables

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0 page-item records.

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Diagrams

Source comparison recorded

3 page-item records. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

$a\ge2$ ; more precisely 3.14619….

Exact private source page references

04 Functions sol.pdf · p.16
04 Functions sol.pdf · p.17
04 Functions.pdf · p.4
04 Functions.pdf · p.9

04-11 · Self-inverse functions alternate by parityQuestion pp.4 / Solution pp.17, 18

Question and explanations → · Complete native source article →

[2020/CJC/Promo/10]. Printed/task labels inventoried: i, ii, iii, iv, v.

Question / subparts

Source comparison recorded

2 page-item records.

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Solution steps

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3 page-item records.

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Teacher notes

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2 page-item records.

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Tables

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0 page-item records.

Inspect native target → I compared this with the original

Diagrams

Source comparison recorded

1 page-item record. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

Inverse requires 2a+bc≠0; its input excludes a/c.

Exact private source page references

04 Functions sol.pdf · p.17
04 Functions sol.pdf · p.18
04 Functions.pdf · p.4
04 Functions.pdf · p.9

04-12 · A logarithmic involutionQuestion pp.5 / Solution pp.19

Question and explanations → · Complete native source article →

[2021/NYJC/Prelim/I/5]. Printed/task labels inventoried: a.i, a.ii, b.i, b.ii.

Question / subparts

Source comparison recorded

2 page-item records.

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Solution steps

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Teacher notes

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3 page-item records.

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Tables

No item inventoried — owner check needed

0 page-item records.

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Diagrams

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0 page-item records.

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Short answer: source-compared
Mathematical correctness: reviewed-with-qualification. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

The source states the transformed x-intercept is the only point. That only identifies the determinable intercept; an unknown y-intercept may exist.

Essential answer conditions and source qualifications

a(i)

A y-intercept (0,f(−3)) may exist but cannot be numerically determined.

Exact private source page references

04 Functions sol.pdf · p.19
04 Functions.pdf · p.5
04 Functions.pdf · p.9

04-13 · A quarter ellipse becomes a self-inverse circleQuestion pp.5 / Solution pp.20, 21

Question and explanations → · Complete native source article →

[2011/ACJC/Prelim/I/9]. Printed/task labels inventoried: sketch, inverse, i, ii.

Question / subparts

Source comparison recorded

2 page-item records.

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Solution steps

Source comparison recorded

3 page-item records.

Inspect native target → I compared this with the original

Teacher notes

Source comparison recorded

1 page-item record.

Inspect native target → I compared this with the original

Tables

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Diagrams

Source comparison recorded

4 page-item records. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

Sketch

$a>0$ , x,y≥0.

Inverse

Strictly decreasing original f is one-one.

Identity on the entire domain forces Df=Rf.

Exact private source page references

04 Functions sol.pdf · p.20
04 Functions sol.pdf · p.21
04 Functions.pdf · p.5
04 Functions.pdf · p.9

04-14 · Recover the unknown function by undoing a logarithmQuestion pp.5 / Solution pp.22, 23

Question and explanations → · Complete native source article →

[2018/DHS/Prelim/I/5]. Printed/task labels inventoried: i, ii, iii.a, iii.b.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

3 page-item records.

Inspect native target → I compared this with the original

Teacher notes

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0 page-item records.

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Tables

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Diagrams

Source comparison recorded

1 page-item record. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

iii(b)

Values outside this interval are not fixed by the stated condition.

Exact private source page references

04 Functions sol.pdf · p.22
04 Functions sol.pdf · p.23
04 Functions.pdf · p.5
04 Functions.pdf · p.9

04-15 · Compose through a restricted range, then use symmetry for areaQuestion pp.6 / Solution pp.23, 24

Question and explanations → · Complete native source article →

[2010/TJC/Prelim/II/3]. Printed/task labels inventoried: a.injectivity, a.composition, b.inverse, b.area.

Question / subparts

Source comparison recorded

2 page-item records.

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Solution steps

Source comparison recorded

3 page-item records.

Inspect native target → I compared this with the original

Teacher notes

Source comparison recorded

1 page-item record.

Inspect native target → I compared this with the original

Tables

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Diagrams

Source comparison recorded

4 page-item records. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed-with-qualification. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Source discrepancy: the question uses ℝ₀⁺ but the solution uses x>0. The inclusion of 2a in the inverse domain awaits clarification.

Essential answer conditions and source qualifications

a — composite

gf exists because Rf avoids −3. fg does not exist because Rg contains inputs≥4.

b — inverse

If ℝ₀⁺ means x≥0: inverse domain (a,2a]. If intended x>0 as solution uses: (a,2a). Endpoint ambiguity remains explicit.

Exact private source page references

04 Functions sol.pdf · p.23
04 Functions sol.pdf · p.24
04 Functions.pdf · p.6
04 Functions.pdf · p.9

04-16 · A non-invertible function can have an invertible compositeQuestion pp.6 / Solution pp.25, 26

Question and explanations → · Complete native source article →

[2011/RI/Promo/13]. Printed/task labels inventoried: a, b.i, b.ii, b.iii, b.iv.

Question / subparts

Source comparison recorded

2 page-item records.

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Solution steps

Source comparison recorded

3 page-item records.

Inspect native target → I compared this with the original

Teacher notes

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Tables

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Diagrams

Source comparison recorded

3 page-item records. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

b(iii)

The alternative factor g(α)+g(β)−6 is positive because both g-values exceed 4.

Exact private source page references

04 Functions sol.pdf · p.25
04 Functions sol.pdf · p.26
04 Functions.pdf · p.6
04 Functions.pdf · p.9

04-17 · Composition conditions and a hidden constant caseQuestion pp.7 / Solution pp.27, 28

Question and explanations → · Complete native source article →

[2019/TJC/Promo/9]. Printed/task labels inventoried: i, ii, iii.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

3 page-item records.

Inspect native target → I compared this with the original

Teacher notes

No item inventoried — owner check needed

0 page-item records.

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Tables

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Diagrams

Source comparison recorded

3 page-item records. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed-with-qualification. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Source qualification: solution PDF p.28 assumes ab≠1 in part (ii). Under that part’s literal wording, a constant-map family also works. Part (iii) requires an inverse, which rules out constant maps; if that assumption is intended throughout, the printed part (ii) answer applies.

Essential answer conditions and source qualifications

The a=1/b constant family satisfies literal (ii). If invertibility in (iii) is intended throughout, retain only a=−1.

iii

Inverse exists only when ab≠1.

Exact private source page references

04 Functions sol.pdf · p.27
04 Functions sol.pdf · p.28
04 Functions.pdf · p.7
04 Functions.pdf · p.9

An empty omission list is not a completeness certificate. Add any issue you find to the article or page-item record before calling that part finished.