A Level · Cambridge (CIE) · 9709

Define a function and find its domain and range

Distinguish permitted inputs from actual outputs, then test whether different inputs can share an output.

Section 1.2 · Functions

Pure Mathematics 1 · Functions · 1.2

Pure Mathematics 1 (Paper 1) · AS & A Level Mathematics 9709, 2026–2027. Written practice is self-checked; show your reasoning and domain restrictions.

Before you begin

Substitute values into a rule and complete a quadratic square.
Understand non-negative square roots and non-zero denominators.

Work on paper. Try each question before revealing help, and explain why your method works.

Quick readiness check

Which input is excluded by the rule 1/(x − 1)?

What you will learn

Explain function, domain, range and one-one using a rule and its stated inputs.
Find ranges for simple quadratic and reciprocal functions, respecting endpoints.

A rule needs its input set

A function assigns exactly one output to each input in its domain. The domain is the set of permitted inputs; the range is the set of outputs actually obtained from those inputs. Writing f: x ↦ x² describes the rule, while a domain such as x ≥ 0 supplies the permitted inputs. The same rule with a different domain can have a different range or different inverse behaviour.

Different inputs may share an output and still form a function: x² sends both 2 and −2 to 4. A one-one function has different outputs for every pair of different permitted inputs. Graphically, a vertical line meets a function graph at most once; a horizontal line meets a one-one function graph at most once. Test one-one on the stated domain rather than on an imagined wider graph.

Find actual outputs, not just a convenient bound

For f(x) = (x − 2)² + 1 on x ≥ 2, the square takes every value from zero upwards, so the range is y ≥ 1. The endpoint 1 is attained at x = 2. On all real x the same range holds, but the function is no longer one-one: inputs equally spaced on opposite sides of 2 share an output.

For g(x) = 3/(x + 2) on x > −2, the denominator takes every positive value, so the range is y > 0. Zero is approached for large x but never attained at any finite input. For a restricted interval, evaluate permitted endpoints and any turning point inside it. An open endpoint or an approached limiting value may be excluded from the range.

Pause and explain

Why is x ↦ x² on all real x a function but not one-one?

Put the idea to work

Worked example

Find the range of g(x) = 3/(x + 2) for x > −2 and decide whether g is one-one.

Show the worked solution

The denominator x + 2 is positive and can take any positive value.
Dividing 3 by those values produces every positive output, but never zero.
If two outputs are equal, 3/(x₁ + 2) = 3/(x₂ + 2) gives x₁ = x₂, so g is one-one.

Answer Range y > 0; g is one-one on x > −2.

Try a topic test

Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.

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From guided practice to a new situation

Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.

1 · Guided

For f(x) = 4 − x² on all real x, find the range and decide whether f is one-one.

Give me a hint

Use x² ≥ 0 and compare inputs on opposite sides of zero.

Compare my reasoning

Since x² ≥ 0, f(x) ≤ 4, with equality at x = 0.
Every y ≤ 4 is attained by a real x satisfying x² = 4 − y.
Inputs 1 and −1 both give 3, so the function is not one-one.

Range y ≤ 4; f is not one-one.

Look for these in your work

Justify that every output in the proposed range is attainable.
Give two distinct inputs with the same output.

2 · Independent

Find the range of h(x) = 1/(x − 1) for x ≥ 3.

Give me a hint

The denominator starts at 2 and can grow without bound.

Compare my reasoning

For x ≥ 3, x − 1 ≥ 2, so 0 < h(x) ≤ 1/2.
The largest output 1/2 is attained at x = 3.
Zero is never attained, while every output between zero and 1/2 has an allowed reciprocal denominator.

0 < y ≤ 1/2.

Look for these in your work

Include the attained maximum.
Exclude the limiting output zero.

3 · Transfer

A supplied sensor model is S(t) = (t − 3)² + 2 for 0 ≤ t ≤ 5. Find its range and decide whether different allowed times can give the same reading.

Give me a hint

Check the vertex and both interval endpoints.

Compare my reasoning

The vertex t = 3 is permitted and gives the minimum S = 2.
The endpoint readings are S(0) = 11 and S(5) = 6; the continuous quadratic attains all readings between 2 and 11 on this interval.
Times 1 and 5 both give 6, so the model is not one-one over the full stated interval.

2 ≤ S ≤ 11; readings can repeat, for example S(1) = S(5) = 6.

Look for these in your work

Check interior turning points as well as endpoints.
Use permitted times when demonstrating a repeated reading.

Common mistakes

Domain describes inputs; range describes actual outputs.
A limiting value is not automatically an included endpoint.
A function can be many-to-one; one-one is an additional condition.

Recall without your notes

What can you explain now?

Find the range of f(x) = (x + 1)² for x ≤ −1. Is f one-one on this domain?

Compare with the explanation

Range y ≥ 0; f is one-one on x ≤ −1.

The allowed inputs use only the left branch of the quadratic, with output zero at −1 and all positive outputs attained once.

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Make it stick

Explain domain, range and one-one to someone using one quadratic and one reciprocal example. Explain why a value approached forever may still be outside the range.

If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.

Keep learning Mathematics

Check the official syllabus 2026–2027

Course scope and assessment

312 sequenced lessons provide 936 guided, independent and application problems covering the 160 objectives in sections 1.1–1.8, 2.1–2.6, 3.1–3.9, 4.1–4.5, 5.1–5.5 and 6.1–6.5. The map lists all 38 sections across six alternative components; all eight Pure Mathematics 1 sections and all six Pure Mathematics 2 sections, from Paper 2 Algebra through Numerical solution of equations, currently have full teaching sequences. Paper 3 Algebra, section 3.1, links the six shared foundation lessons plus six lessons on partial fractions and rational binomial expansions; Paper 3 logarithms, trigonometry and numerical equations use the shared lessons, with the Paper 1 graph foundation linked into trigonometry. Paper 3 differentiation combines the shared methods with its inverse-tangent derivative lesson. Paper 3 Integration links four shared standard and identity lessons plus seven advanced lessons, covering all six section 3.5 requirements. Eight Vectors lessons cover all six section 3.7 requirements, with 2D and 3D line and solid applications. Nine Differential equations lessons cover all four section 3.8 requirements, including Paper 3 integration techniques, initial conditions, equilibrium solutions and conditional model interpretations. Eleven Complex numbers lessons cover all eight section 3.9 requirements, with seven Argand diagrams and explicit argument and boundary checks. All nine Pure Mathematics 3 sections now have teaching linked to every mapped requirement. Nine Forces and equilibrium lessons cover all seven section 4.1 requirements, with six force diagrams, signed incline-friction intervals, reaction-dependent pull and push limits and separate-body third-law checks. Nine Kinematics lessons cover the four official section 4.2 requirements, with the graph requirement mapped to three relationships. Six motion diagrams support directed quantities, graph slopes and signed areas, Paper 1 time-calculus methods, reversal distances, constant-acceleration stages and shared-clock particle meetings. Six Momentum lessons cover both section 4.3 requirements, including signed system totals, conservation assumptions, outgoing velocities, coalescence, unknown masses and successive impacts. Four diagrams show system momentum through the stipulated events, and supplied kinetic-energy checks distinguish conservation of momentum from changes in kinetic energy. Twelve Newton’s laws lessons cover all four section 4.4 requirements, with vertical support and release stages, rough horizontal and incline motion, inferred parameters, hanging and surface pulley systems, rigid tow-bar thrust and slack-string transitions. Seven force diagrams support the separate-particle balances. Nine Energy, work and power lessons cover all five section 4.5 requirements, with angle-dependent work, kinetic and reference potential energy, signed net-work balances, smooth curved slides, rough and driven slopes, ideal connected-system energy and instantaneous power on hills. Five diagrams distinguish the energy inputs, losses and valid positive-speed force model. All five Mechanics sections now have teaching and staged practice linked to their mapped requirements. Thirteen Representation of data lessons cover all five section 5.1 requirements, including choice and limitations of displays, ordered and back-to-back stems, raw summaries and box plots, unequal-width density histograms, cumulative-frequency interpolation, descriptive moment calculations, grouped estimates, coding, combining two datasets and correcting totals. Six accessible diagrams retain repeated observations and distinguish numerical spread, frequency area and estimated readings. Ten Permutations and combinations lessons cover both official section 5.2 requirements, with selection and ordered roles, repeated objects, adjacency blocks, non-adjacency gaps, leading-zero cases, labelled multi-row seating and mixed restrictions. Six accessible diagrams show outcome trees, ordered versus unordered choices, repeated copies, blocks, gaps and distinct row seats. Eleven Probability lessons cover all four section 5.3 requirements, with equiprobable elementary outcomes, selection-based probabilities, complements and disjoint regions, independence tests, restricted sample spaces, two-way tables, replacement and removal trees, reversed conditions and unknown parameters. Seven accessible diagrams show dice outcomes, random subsets, event regions, a conditioned grid, a count table and two probability trees. Twelve Discrete random variables lessons cover all three official section 5.4 requirements, with complete probability tables, moment and parameter checks, binomial assumptions, exact probabilities and moments, geometric first-success probabilities, tails, expectation and integer reliability thresholds. Six accessible diagrams distinguish outcome grouping, weighted moments, fixed trials, discrete probability bars, waiting conventions and integer tail deadlines. Twelve Normal distribution lessons cover all three official section 5.5 requirements, with continuous-area interpretation, MF19 cumulative tables, full standardisation, tail and interval regions, conditional probabilities, percentiles, one- and two-parameter inference, binomial suitability, continuity corrections and approximate integer thresholds. Seven accessible diagrams show model areas, table reading, aligned scales, intervals, quantiles, matched binomial and normal shapes and whole integer cells. All five Statistics 1 sections now have teaching and staged practice linked to their mapped requirements. Fourteen Poisson distribution lessons cover all five official section 6.1 requirements, with homogeneous event-count assumptions, exposure scaling, exact masses and tails, conditional count events, moments, parameter inference, probability duration targets, disjoint and overlapping windows, rare-count binomial approximations, matched normal approximations and continuity corrections. Nine accessible diagrams show event exposures, unit conversions, discrete bars, complete tail complements, population moments, shared windows, approximate count models and integer cells. Exact checks distinguish approximate and exact close-target decisions. Fourteen Linear combinations lessons cover the six listed results in official section 6.2, with affine moments, signed scales and constant cases, expectation without independence, independent squared variance contributions, fixed and random fees, independent copies versus reused values, affine normal events, independent normal sums and differences, weighted and conditional areas, coefficient inference, exact independent Poisson sums and overlapping exposure limits. Nine accessible diagrams distinguish scale changes, variance contributions, component costs, copy mechanisms, aligned normal scales, summed areas, Poisson count pairs and unsupported distribution claims. Fourteen Continuous random variables lessons cover both official section 6.3 requirements, with density validity and units, support-aware intervals, complements and conditional regions, first and second moments, percentiles by direct density areas, symmetry checks, parameter inference, infinite-support power and exponential integrals, divergent-moment limits and combined applications. Nine accessible diagrams distinguish density height from probability, normalisation, negative-density failures, central and conditional areas, mean versus median, percentile locations and partial infinite tails. Explicit cumulative distribution function knowledge is not required. Eighteen Sampling and estimation lessons cover all eight official section 6.4 requirements, with population and frame definitions, random-number selection, sampling-quality limits, independent sample-mean moments, normal means, CLT conditions, SE scaling, raw, summarised and coded unbiased estimates, known-variance and large-sample mean intervals, repeated-sampling interpretation, precision planning, proportion intervals, sparse-outcome limits and contextual applications. Ten accessible diagrams distinguish selection frames, number rejection, repeated sample means, population versus mean density, CLT models, variance divisors, interval targets, illustrative coverage, integer size checks and proportion margins. Eighteen Hypothesis tests lessons cover all five official section 6.5 requirements, with pre-specified alternatives, inclusive count tails, explicit equal-tail conventions, integer critical regions and actual significance, exposure-aware Poisson tests, continuity-corrected count approximations, normal and large-sample mean tests, confidence-interval connections, error classifications, fixed-region binomial, Poisson and normal error probabilities and sample-size power comparisons. Nine accessible diagrams show the testing procedure, exact count regions, unequal Poisson tails, discrete significance jumps, corrected count boundaries, matching mean intervals and alternative error distributions. All shared lessons are counted once in the course total. All 38 sections now link teaching and practice. Eight earlier overview lessons remain available, but do not count as full section coverage. This is a content-availability measure, not full programme completion or exam readiness. Written practice is self-checked, not officially marked. Check the 2028–2030 syllabus separately if sitting an examination from 2028; this teaching map uses 2026–2027 requirements.