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Worked solution

Choose an injective restriction without hiding ambiguity

Given that f:x↦x2f:x\mapsto x^2, x∈Rx\in\mathbb R, −2≤x≤3-2\le x\le3. Show that ff is not one-to-one. A function gg is a restriction on ff; state the largest possible domain of gg, for which gg is one-to-one.

CJC · Functions · Example 5

Show that f is not one-to-one

Answer

f is not one-to-one.

The Teacher draws the full restricted parabola with included endpoints (−2,4)(-2,4) and (3,9)(3,9), and the line y=1y=1. That line meets it twice: independently f(−1)=f(1)=1f(-1)=f(1)=1 with two distinct permitted inputs. One counterexample disproves injectivity.

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Parabola y=x² on[−2,3], included endpoints (−2,4),(3,9), vertex (0,0), horizontal Teacher line y=1 meeting it at −1 and 1.
Parabola y=x² on[−2,3], included endpoints (−2,4),(3,9), vertex (0,0), horizontal Teacher line y=1 meeting it at −1 and 1.
Parabola y=x² on[−2,3], included endpoints (−2,4),(3,9), vertex (0,0), horizontal Teacher line y=1 meeting it at −1 and 1.
How did this part go?

State the source restriction and its qualification

Answer

The Teacher selects Dg=[0,3]D_g=[0,3]. This is the longest interval restriction; “largest” without a criterion is ambiguous.

On [0,3][0,3], squaring is strictly increasing, so the Teacher’s selected restriction is one-to-one. Independently the other inclusion-maximal interval is [−2,0][-2,0], where squaring is strictly decreasing. Any interval containing both a negative and a positive input contains a small pair ±t\pm t and is not injective. Thus [0,3][0,3] has the greater interval length, 3 rather than 2. Under set inclusion neither of those two maximal intervals contains the other, so no unique greatest injective domain exists. If arbitrary non-interval subsets are permitted, further maximal choices can select one sign from each pair. Keep the printed ask and answer with this qualification rather than silently adding an interval/length condition to the prompt.

How did this part go?

Solution notes

The ambiguity was already recorded in evidence/cjc-functions-inverse-review.md; this import retains it explicitly. No printed marks or external exam attribution.