Functions
Welcome to Functions — the heart of algebra and the widest area of the CSCA maths syllabus. A function is like a machine: you feed it an input, it applies a rule, and it gives you an output. Understanding functions means understanding how things change.
In this module, you will learn to describe functions through their domain and range, handle piecewise definitions, determine monotonicity (单调性) and parity (奇偶性), find inverses and compositions, and then dive deep into exponential and logarithmic functions — the tools that model everything from population growth to radioactive decay. You will then master transformations that shift, stretch, and reflect graphs, learn the general term and sum of arithmetic and geometric sequences, and finally meet power functions and the trigonometric functions.
Let's get started!
What the Exam Tests
The syllabus's Functions area lists the concepts and properties of functions (domain, range, monotonicity, parity), the basic elementary functions (power, exponential, logarithmic and trigonometric), sequences (general terms and sums of arithmetic and geometric sequences) and the basics of derivatives. This module teaches all of it except derivatives, which are in Calculus & Statistics.
Questions can ask you to read a property from a formula or a graph, compare values such as log₂3 and 2^0.5, solve an exponential or logarithmic equation, find the period or range of a trigonometric function, or find a term or a sum of a sequence.
In our mock papers the Functions area, with the derivative and integration lessons, gets 16 of the 48 questions. The CSCA publishes the topics but not how many questions each gets, so this split is our own reading of the syllabus.
The domain (定义域) of a function f is the set of all valid inputs (x-values). The range (值域) is the set of all possible outputs (y-values or f(x)-values).
Finding the Domain
Ask: "What values of x would cause a problem?" Common restrictions:
- Division by zero: For f(x) = 1/(x-3), exclude x = 3. Domain: ℝ \ {3} or (-∞,3) ∪ (3,∞).
- Square root of negative: For f(x) = √(x-2), need x-2 ≥ 0 → x ≥ 2. Domain: [2, ∞).
- Logarithm of non-positive: For f(x) = ln(x), need x > 0. Domain: (0, ∞).
- Combined: For f(x) = √(x) / (x-4), need x ≥ 0 AND x ≠ 4. Domain: [0,4) ∪ (4,∞).
Finding the Range
Methods include:
- Algebraic: Set y = f(x), solve for x in terms of y, and find what y-values are valid.
- Graphical: Sketch or visualise the function and read off the y-values covered.
- Key observations: x² ≥ 0 always, so x² + 3 ≥ 3. Range of f(x) = x² + 3 is [3, ∞).
Notation: Domain and range are sets, so express them using set or interval notation.
Think of the domain as the "menu" of a restaurant (what you can order) and the range as the "dishes that actually come out of the kitchen" (what you actually get).
Key takeaway: Domain = all valid inputs (avoid ÷0, √negative, log of non-positive). Range = all possible outputs. Express both in interval notation.
Key Formulas
√(expr) needs expr ≥ 0 | 1/expr needs expr ≠ 0 | log(expr) needs expr > 0
Worked Example 1
Example 1: Find the domain of f(x) = √(6 - 2x).
Need 6 - 2x ≥ 0 → -2x ≥ -6 → x ≤ 3. Domain: (-∞, 3].
Worked Example 2
Example 2: Find the domain and range of f(x) = 1/(x² + 1).
Domain: x² + 1 ≥ 1 > 0 for all x, so no restrictions. Domain = ℝ.
Range: Max value is 1/(0+1) = 1 (at x=0). As x → ±∞, f → 0⁺. Range = (0, 1].
Worked Example 3
Example 3: Find the domain of g(x) = ln(x² - 4).
Need x² - 4 > 0 → x² > 4 → |x| > 2 → x < -2 or x > 2.
Domain: (-∞, -2) ∪ (2, ∞).
Exam tip: For domain, systematically check for division by zero, square roots, and logarithms.
Exam tip: For range, try the algebraic method: set y = f(x), solve for x, and see which y-values produce real x.
Common mistake: Forgetting to exclude log argument = 0: ln(0) is undefined, so you need strict inequality x > 0, not x ≥ 0.
Common mistake: Saying range of x² is ℝ: x² ≥ 0 always, so range is [0, ∞), not all reals.
Try This Yourself
Find the domain of f(x) = √(x+3) / (x-1).
Module Summary
In this module you mastered functions from the ground up: domain and range, piecewise definitions, symmetry, injectivity, inverses, and compositions. You then explored the world of exponentials and logarithms — from basic properties to growth/decay models and equation-solving techniques. Finally, you learned to transform any function graph through shifts, stretches, and reflections. You also covered power functions, the trigonometric functions and their graphs, and arithmetic and geometric sequences, including the nth term and the sum of the first n terms (and, as an extension, when an infinite geometric series converges). These skills are foundational for calculus and for much of the CSCA syllabus.
Open and read all sections to complete this module