Mathematical Methods
Functions, calculus, probability — tech-free and extended-response exams with fully worked solutions.
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Functions, Domains & Function Notation
What Is a Function?
A function is a relation between two sets — the domain (set of allowable inputs) and the codomain (set of possible outputs) — where every element of the domain is mapped to exactly one element of the codomain. This is the defining property: one input, one output. No x-value may be paired with two different y-values.
We write f : A → B, read "f is a function from A to B", where A is the domain and B is the codomain. The rule of the function tells us what to do to each input. For example, f : ℝ → ℝ, f(x) = x² maps every real number to its square.
Distinguishing functions from non-functions: The vertical line test is the graphical check — a relation is a function if and only if every vertical line drawn through the graph intersects it at most once. For instance, the circle x² + y² = 4 is a relation but not a function, because the vertical line x = 0 intersects it at (0, 2) and (0, −2).
Worked Example: Is y² = x a function?
Choose x = 4. Then y² = 4, so y = 2 or y = −2. The input x = 4 produces two outputs, so this is not a function. ✓
By contrast, y = x² is a function: for each x, x² produces exactly one value. The vertical line test confirms: every vertical line hits the parabola at most once. ✓
Note: in Methods, functions are typically real-valued and defined on real-number domains. The notation f(x) denotes the output (or image) of x under f, and is read "f of x".
Domain, Codomain and Range
The domain of a function is the complete set of input values for which the rule is defined and produces a real output. The range (also called the image) is the actual set of output values produced — it is always a subset of the codomain.
Maximal domain: when no domain is stated, we assume the maximal (natural) domain — the largest subset of ℝ for which the rule yields a real number. Two key restrictions apply:
- You cannot divide by zero: exclude any x that makes a denominator equal zero.
- You cannot take the square root (or any even root) of a negative number (in the reals): exclude any x that makes the expression under the radical negative.
- You cannot take the logarithm of zero or a negative number: for loga(g(x)), require g(x) > 0.
Worked Example 1 — Rational function: Find the maximal domain of f(x) = 3/(x − 2).
Denominator: x − 2 ≠ 0 ⟹ x ≠ 2.
Maximal domain: ℝ \ {2} = (−∞, 2) ∪ (2, +∞). ✓
Worked Example 2 — Square root function: Find the maximal domain and range of f(x) = √(9 − x²).
Require 9 − x² ≥ 0 ⟹ x² ≤ 9 ⟹ −3 ≤ x ≤ 3.
Maximal domain: [−3, 3].
Now find the range: √(9 − x²) ≥ 0 always, and the maximum occurs at x = 0: √9 = 3. As x → ±3, √(9 − x²) → 0.
Range: [0, 3]. ✓ (Geometrically, this is the upper semicircle of radius 3.)
Worked Example 3 — Logarithmic function: Find the maximal domain of f(x) = loge(2x − 6).
Require 2x − 6 > 0 ⟹ 2x > 6 ⟹ x > 3.
Maximal domain: (3, +∞). ✓
Range: loge of a positive quantity ranges over all of ℝ, so range = ℝ.
When a domain is specified, the range must be determined by examining the function's behaviour (including turning points, endpoints, and asymptotes) over that restricted domain — not the maximal domain.
The graph of y = f(x) passes through the point (2, 5). The graph is transformed to y = 2f(x − 1) + 3. The image of the point (2, 5) under this transformation is
- (1, 7)
- (3, 13)
- (3, 8)
- (1, 13)
Show the worked answer
Answer: B
x-coordinate: 2 + 1 = 3. y-coordinate: 2(5) + 3 = 13. Image = (3, 13).
All 20 practice exams
- Exam 1 — Calculus applied to a logistic population-growth model: rates of change, optimisation of growth rate, average value and definite integrals; Functions, transformations and inverse functions arising from exponential/logistic models, with domain and range; Algebra: index and log laws, solving exponential and logarithmic equations exactly over a domain
- Exam 2 — Calculus: differentiation rules, definite integrals, optimisation and stationary points applied to projectile motion; Algebra: solving exponential, logarithmic and trigonometric equations over a domain; Functions: quadratic and circular models, transformations, domain/range
- Exam 3 — Calculus of exponential decay models (derivatives, rates, tangents, antidifferentiation, definite integrals/AUC, average value); Exponential and logarithmic algebra: index/log laws and solving equations exactly over a domain; Probability: discrete random variables, binomial distribution, continuous pdfs and the normal distribution
- Exam 4 — Circular (trig) functions: modelling periodic tidal motion with sinusoids — amplitude, period, mean, range, and solving trig equations over a domain; Calculus applied to trig models: derivatives, rates of change, slack water (stationary points), fastest rise/fall (points of inflection), average value via definite integrals; Probability & statistics in context: binomial distribution (mean/variance), the normal distribution (z-scores), and approximate 95% confidence intervals for a population proportion
- Exam 5 — Calculus: differentiation (product/chain rules, derivatives of e^x and log), antidifferentiation, definite integrals, area between curves, average value, and optimisation via stationary points; Algebra: solving exponential equations (quadratic-in-disguise), circular equations over a domain, literal/inverse functions and tangents; Probability & statistics: binomial distribution (E(X), Var(X)), the normal distribution (z-scores), and approximate confidence intervals for a population proportion
- Exam 6 — Calculus: differentiation of exponential models, tangents, rates of change, antidifferentiation and definite integrals (area, average value) in technology-free form; Algebra: index and log laws, solving exponential and logarithmic equations exactly over a domain, half-life modelling; Functions: exponential and log functions, transformations, inverse and composite functions, asymptotes and domain/range
- Exam 7 — Binomial distribution in a quality-control context: exact by-hand probabilities (Section A) and full CAS modelling with E(X), Var(X), conditional probability and minimum-sample-size (Section C); Sample proportions and approximate 95%/90% confidence intervals for a population proportion of defectives, including margin-of-error sample-size design; Continuous random variables: constructing a probability density function, mean, median, variance and interval probabilities for a manufacturing dimension
- Exam 8 — The normal distribution: probabilities, conditional probability, quantiles/inverse, and standardisation (z-scores); Sample proportions and approximate confidence intervals for a population proportion; Calculus modelling: differentiation, stationary points, average value and definite integrals applied to context
- Exam 9 — Sample proportions and approximate confidence intervals for a population proportion; Binomial distribution (mean np, variance np(1-p)) and the normal approximation; Calculus: differentiation, stationary points and rates of change applied to a modelling context
- Exam 10 — Calculus: differentiation (product/chain), antidifferentiation & definite integrals, stationary points and their nature, average value; Algebra: solving exponential equations (quadratic-in-e^x), trigonometric equations over a domain, inverse functions with domain/range; Probability & statistics: binomial distribution, continuous PDFs, the normal distribution and approximate confidence intervals for a proportion
- Exam 11 — Circular function modelling: differentiation, max rate, period/range and solving trig equations over a domain (Ferris wheel height); Calculus applications: tangents, average value via definite integral, optimisation and area under exponential models; Probability & statistics: binomial occupancy, normal distribution z-scores, sample proportion confidence intervals and continuous pdfs
- Exam 12 — Exponential and logarithmic models for compound growth (index/log laws, e^x, solving exponential and log equations technology-free); Calculus of growth models: differentiation of exponential/product forms, rates of change, antidifferentiation and definite integrals; Financial modelling with first-order recurrence relations (reducing-balance loans, annuities) and continuous vs periodic compounding using CAS
- Exam 13 — Calculus in context: derivatives, rates of change, stationary/inflection points and definite-integral measures (average value, total change) applied to exponential and logistic growth models; Exponential and logarithmic algebra: index/log laws, solving exponential & log equations, inverse/composite functions and transformations (technology-free); Probability & statistics: binomial distribution, the normal distribution (z-scores, percentiles) and approximate confidence intervals for a population proportion
- Exam 14 — Calculus: rates of change, related rates, optimisation and definite integrals applied to a water-tank context; Functions and graphs: quadratic, exponential and circular models with domain/range and asymptotic behaviour; Technology-free differentiation (product/chain rule, first principles) and exact-value solving of log/exponential/trig equations
- Exam 15 — Normal distribution & tolerances; Binomial distribution; Sample proportions & confidence intervals
- Exam 16 — Calculus in a projectile context: differentiation, tangents/first principles, antidifferentiation, definite integrals, average value and optimisation; Functions and algebra: exponential/log/trig equations over a domain, inverse and composite functions, hybrid-function continuity and differentiability; Probability and statistics: binomial distribution, the normal distribution with z-scores, and confidence intervals for a population proportion
- Exam 17 — Calculus in context (rates of change, average value, optimisation) applied to a sinusoidal temperature model; Circular functions: amplitude/period/midline, exact-value equation solving, and transformations; Probability and statistics: normal distribution, binomial, and confidence intervals for a population proportion
- Exam 18 — Differential & integral calculus (product rule, stationary points, antidifferentiation, definite integrals, average value, optimisation); Probability distributions (binomial E(X)/Var(X), continuous pdfs, the normal distribution, conditional probability); Statistical inference (sample proportions and approximate confidence intervals for a population proportion)
- Exam 19 — Differential calculus: product/chain/quotient rules, stationary points and nature, tangents, rates of change; Integral calculus: antidifferentiation, definite integrals, area, average value of a function; Probability: binomial distribution, the normal distribution, continuous random variables and probability density functions
- Exam 20 — Sample proportions and approximate confidence intervals for a population proportion; Binomial distribution: exact probabilities, mean np and variance np(1-p), conditional probability; The normal distribution: z-scores, percentiles, and the sampling distribution of a sample proportion
All 20 revision notes
- Functions, Domains & Function Notation
- Solving Equations & Simultaneous Systems
- Technology-Free (Exam 1) Skills
- Antidifferentiation & Definite Integrals
- Applications of Differentiation
- Differentiation: Rules & Techniques
- Circular (Trigonometric) Functions
- Inverse & Composite Functions
- Polynomial Functions & Their Graphs
- Power, Exponential & Logarithmic Functions
- Transformations of Functions & Graphs
- Further Integration & Applications
- Rates of Change & Modelling with Calculus
- Combining Functions in Calculus Contexts
- Continuous Random Variables & PDFs
- Discrete Random Variables
- Modelling & Problem Solving
- Sample Proportions & Confidence Intervals
- The Binomial Distribution
- The Normal Distribution