IEMJEE Maths preparation: One-to-one functions — worked example and reasoning

Problem to solve

Is f(x)=x² injective on all real numbers?

Try the problem before reading the answer. Record the requested quantity, relevant information and any interpretation choices. Keep this original practice example separate from official examination questions or previous-year papers.

Choose the governing principle

Test whether f(a)=f(b) forces a=b within the stated domain.

An injective function assigns different outputs to different allowed inputs. The choice of domain matters.

Worked answer

No: f(2)=f(−2)=4. It becomes injective on the restricted domain x≥0.

The answer follows the stated model and the values in this question. Preserve exact expressions when given; if you use a numerical approximation, label it and keep sufficient precision until the final step.

Audit the reasoning

Substitute a candidate result into the original statement where possible. Inspect sign, endpoint inclusion and whether a real or complex answer is requested. The simplified expression and the original expression can have different domains even when their nearby values agree.

A specific error to avoid here: A graph passing the vertical-line test is a function, but it need not be one-to-one.

Explain before repeating

Write a short explanation linking the governing principle to the answer rather than copying the final line. Identify one assumption and explain how violating it would make this method unreliable. Use the linked worksheet for a new task, then return to this example after a delay and solve it without the answer visible.

Official sources and verification

Sources checked on 26 October 2026. A source’s date, programme and campus limit which claims it supports. Official pages may change after review.

1. UEM — IEMJEE examination information

   Historical examination pattern; verify the current cycle on the central portal.

2. NTA — explicitly labelled JEE Main 2026 syllabus

   Historical PCM syllabus reference. Lessons and exercises are independently authored, not official IEMJEE papers or verified 2027 paper predictions.

3. NCERT — official textbook catalogue

   Primary textbook reference for checking school-level concepts. This directory does not certify the independent lesson or establish IEM examination weightage.

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