H2 Maths Functions and Graphs | Free Notes

Q: What does H2 Maths Notes (JC 1-2): 1) Functions and Graphs cover?
A: Detailed guide to MOE Topic 1 -- functions, transformations, equations, and inequalities -- with worked examples and calculator workflows.

Before you begin\ Make sure your Additional Maths foundations are fresh (logarithms, surds, quadratic inequalities). Keep the graphing calculator (GC) in Exam Mode while you revise so keystrokes become automatic.

• Functions define inputs; graphs show behaviour: State domain and key features.
• Most marks come from restrictions, transformations, and intervals: Track what changes and what stays fixed.
• A complete solution combines algebra, sketch, and GC evidence where allowed: Write the method before the final answer.

Concrete example: Before finding an inverse for a quadratic, restrict its domain so it is one-to-one. Without that restriction, the inverse is not a function.

For the full topic map, paper weightings, and official PDF link, see our H2 Maths Syllabus 2026-27 overview.

Status: SEAB's current H2 Mathematics (9758) syllabus PDF is labelled for 2026. Topic 1 coverage is within Section A Pure Mathematics, which is assessed in Paper 1 (100 marks) and Paper 2 Section A (40 marks).

1.1 | Functions

Core ideas

• A function maps each input in its domain to exactly one output in its range.
• Injective (one-to-one) functions pass the horizontal-line test and admit inverses on their natural domain.
• Composition $(f \circ g)(x) = f\bigl(g(x)\bigr)$ requires the range of $g$ to sit within the domain of $f$

Domain/range checks

1. State the natural domain (avoid zero denominators, even roots of negatives, logarithms of non-positive numbers).
2. Restrict the domain so the function becomes one-to-one before attempting to invert it.

Example 1 -- Restricting domain for an inverse

Consider $f(x) = 2x^2 - 3x + 4$.

1. $f'(x) = 4x - 3$ has a stationary point at $x = \tfrac{3}{4}$

Composition pitfalls

• Always record the intermediate range when composing functions.
• Do not rely on the shortcut $(f \circ g)^{-1} = g^{-1} \circ f^{-1}$

Composition domain checkpoint

Before finding $(f \circ g)(x)$, check both the input restriction for $g$ and the output restriction needed by $f$. The common mistake is to simplify the formula first and forget that $g(x)$ must land inside the domain of $f$

Worked check: let $f(x)=\sqrt{x}$ and $g(x)=\dfrac{1}{x-2}$

1.2 | Graphs and Transformations

Essential graph features

• Axis intercepts come from solving $f(x) = 0$ and evaluating $f(0)$.
• Stationary points satisfy $f'(x) = 0$

Transformation cheat sheet

Parametric sketches

• For $x = g(t)$ and $y = h(t)$, tabulate $t$ values, show orientation, and eliminate $t$ if possible.
• State domain restrictions arising from both $g(t)$

Example 2 -- Transformation workflow

Sketch $y = -2 f\bigl( \tfrac{x}{3} \bigr) + 1$ given $y = f(x)$

1. Start with the original graph.
2. Stretch horizontally by factor 3 (replace $x$ with $\tfrac{x}{3}$).
3. Stretch vertically by factor 2 and reflect across the x-axis (multiply by $-2$).
4. Translate the result up by 1 unit.

1.3 | Equations and Inequalities

General equation strategy

1. Formulate the equation from the context.
2. Solve exactly when possible; otherwise document the GC feature used (root finder, table, intersection).

Systems of linear equations

• Use row reduction or the GC rref solver.
• Describe the solution set explicitly (single point, family of solutions, or inconsistent).

Inequalities

• Linear or quadratic forms: rewrite as $f(x) > 0$ and use sign diagrams or a sketch.
• Absolute values: rewrite as compound inequalities, e.g. $\lvert x - a \rvert \lt b$ implies $a - b \lt x \lt a + b$

Example 3 -- Rational inequality

Solve $\dfrac{x - 2}{x + 1} > 0$.

1. Critical points: $x = -1$ (vertical asymptote) and $x = 2$ (zero).
2. Test intervals $(-\infty, -1)$, $(-1, 2)$

Example 4 -- Absolute inequality

Solve $\lvert 2x - 3 \rvert \leq 5$.

• Convert to $-5 \leq 2x - 3 \leq 5$.
• Hence $-1 \leq x \leq 4$.

Graphical method with GC

1. Plot $y = f(x)$.
2. Use the intersection tool to locate roots or boundary points.
3. Describe where the graph lies above or below the x-axis.

Practice Quiz

Measure your command of domains, compositions, and inequality workflows before moving on.

1.4 | Quick Revision Checklist

• State domains and ranges confidently, including logarithmic and rational restrictions.
• Describe every transformation's effect on intercepts, asymptotes, and symmetry.
• Solve linear, quadratic, and simple rational inequalities without relying solely on technology.
• Record the calculator feature used (e.g. ISCT, TABLE, Graph solve).

Stay on the H2 Maths notes hub to keep the sequence intact and jump into Topic 2 - Sequences & Series once you nail these fundamentals.

Want weekly guided practice on Functions and Graphs? Our H2 Maths tuition programme builds fluency in this topic through structured problem sets and exam-style drills.

Sources

• SEAB H2 Mathematics syllabus (9758), examinations from 2026 - Topic 1 Functions and graphs (sub-topics 1.1-1.3 covering functions/inverses/composition, graph sketching and transformations, equations and inequalities) under Section A Pure Mathematics. The official PDF link is recorded in the front matter citation for this note.