Functions and graphs open the H2 Mathematics syllabus and underpin almost everything that follows — inverse functions, composite functions and graph transformations are the language the rest of the course is written in. Students who treat this topic as "easy revision" from O-Level often lose marks on the precise conditions H2 demands. This guide is from Ancourage Academy, whose JC H2 Mathematics tuition builds these foundations in small groups of 3–6 at Bishan and Woodlands.
This is a single-topic deep-dive — a sibling to our H2 Math calculus and vectors guides, and part of our wider H2 Mathematics overview. If you did not take A-Maths, first read whether you need A-Maths for H2 Math.
If functions are where the H2 syllabus first felt abstract, Ancourage Academy's JC1 H2 Mathematics programme rebuilds the topic from the definitions — book a trial class (usually $18) for a diagnostic assessment.
What Does Functions & Graphs Cover in H2 Math?
In H2 Mathematics (9758), the functions and graphs strand covers the definition of a function, domain and range, one-one functions and their inverses, composite functions, graphing techniques with the graphing calculator, graph transformations, equations and inequalities, and simple parametric equations. The SEAB Mathematics syllabus (9758) defines exactly what is examinable, and the graphing calculator is assumed throughout.
What Are Domain and Range?
The domain is the set of allowed inputs of a function and the range is the set of outputs it actually produces — and in H2 Math, stating them correctly (in set or interval notation) is itself worth marks.
Two habits separate strong scripts: always restrict the domain to where the rule is defined (for example, excluding values that make a denominator zero or a square root negative), and read the range off the graph rather than guessing. The graphing calculator is the fastest way to confirm a range, but you must still express it precisely.
When Does an Inverse Function Exist?
A function has an inverse if and only if it is one-one — that is, each output comes from exactly one input, which you test with the horizontal line test.
- Existence: f⁻¹ exists only when f is one-one. If f is not one-one, you restrict its domain until it is.
- Domain and range swap: the domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f.
- Graph: the graph of y = f⁻¹(x) is the reflection of y = f(x) in the line y = x.
Because of that reflection, when an increasing function meets its inverse, the intersection lies on the line y = x — so solving f(x) = f⁻¹(x) often reduces to solving f(x) = x. State the condition for the inverse to exist before finding it; omitting it is a common mark-loss point.
How Do Composite Functions Work?
The composite function fg means "do g first, then f" — fg(x) = f(g(x)) — and it exists only when the range of g is contained in the domain of f.
Two points are tested almost every year: the existence condition (range of g ⊆ domain of f), and the fact that composition is not commutative, so fg and gf are generally different functions. When asked for the range of a composite, work outward: find the range of the inner function first, then apply the outer function to that set.
What Graphing Techniques Does H2 Math Expect?
You are expected to sketch graphs showing all key features — intercepts, asymptotes, stationary points and symmetry — and to use the graphing calculator to confirm them.
| Feature | How to find it |
|---|---|
| Axis intercepts | Set x = 0 and y = 0 |
| Vertical asymptote | Where the denominator is zero (after cancelling any common factors) |
| Horizontal / oblique asymptote | Behaviour as x → ±∞ (divide out / long division) |
| Stationary points | Where dy/dx = 0 |
For rational functions, dividing the numerator by the denominator reveals an oblique asymptote that the graphing calculator alone will not label. A clear, labelled sketch is what earns the method marks.
How Do Graph Transformations Work?
A small set of transformations maps y = f(x) onto related graphs, and the modulus graphs y = |f(x)| and y = f(|x|) are the ones students most often confuse.
| Transformation | Effect on y = f(x) |
|---|---|
| y = f(x) + a | Translation a units in the y-direction |
| y = f(x + a) | Translation a units in the negative x-direction |
| y = a f(x) | Stretch parallel to the y-axis by factor |a|; if a < 0, also reflect in the x-axis |
| y = f(ax) | Stretch parallel to the x-axis by factor 1/|a|; if a < 0, also reflect in the y-axis |
| y = |f(x)| | Reflect the parts below the x-axis up above it |
| y = f(|x|) | Reflect the x ≥ 0 part across the y-axis (even graph) |
When several transformations combine, the order matters — apply them in the correct sequence and check one point to confirm. The reciprocal graph y = 1/f(x) is also tested: zeros of f become vertical asymptotes, while non-zero local maxima and minima of f swap into local minima and maxima of 1/f.
The Most Common Functions & Graphs Mistakes
In our H2 Math classes at Ancourage Academy, a handful of recurring errors cause most avoidable mark loss in this topic.
| Mistake | Why it happens | How to fix it |
|---|---|---|
| Finding f⁻¹ without stating the condition | Skipping the one-one check | State that f is one-one (or restrict the domain) before inverting |
| Ignoring the composite existence condition | Forgetting range of g ⊆ domain of f | Check the inner range fits the outer domain first |
| Confusing |f(x)| with f(|x|) | Treating the two modulus graphs alike | |f(x)| reflects up; f(|x|) reflects the right half across the y-axis |
| Range left in the wrong notation | Writing a range as an equation | Use set or interval notation read off the graph |
| Wrong transformation order | Combining stretches and translations carelessly | Apply transformations in sequence and verify with one point |
How Does This Topic Connect to the Rest of H2 Math?
Functions and graphs are the foundation the rest of H2 Mathematics rests on.
- Calculus: stationary points, asymptotes and curve sketching are calculus applied to graphs. See our calculus deep-dive.
- Sequences and series: graphs of functions help visualise convergence and behaviour. See our sequences and series guide.
- Differential equations: families of solution curves are graph sketches. See our differential equations guide.
A Study Plan for Mastering H2 Functions & Graphs
Work this topic in order: definitions first, then inverses, then composites, then graphing and transformations.
- Week 1 — definitions: drill domain and range in correct notation, including restricted domains.
- Week 2 — inverses: master the one-one condition, the domain/range swap, and the reflection in y = x.
- Week 3 — composites: practise the existence condition and finding ranges of composite functions.
- Week 4 — graphing and transformations: sketch with all key features and drill the modulus and transformation graphs.
Ancourage Academy's JC1 and JC2 H2 Mathematics programmes work through functions and graphs on exactly this progression in small groups of 3–6. Book a trial class (usually $18) for a diagnostic, or WhatsApp us with any questions.
Common Questions About H2 Math Functions & Graphs
When does a function have an inverse in H2 Math?
A function has an inverse if and only if it is one-one — each output corresponds to exactly one input, which you confirm with the horizontal line test. If the function is not one-one, you restrict its domain until it is. The domain of f⁻¹ equals the range of f, the range of f⁻¹ equals the domain of f, and the graph of f⁻¹ is the reflection of f in the line y = x.
What is the condition for a composite function fg to exist?
The composite fg(x) = f(g(x)) exists only when the range of the inner function g is a subset of the domain of the outer function f. Always check this condition before forming the composite. Remember that composition is not commutative, so fg and gf are generally different functions, and to find the range of fg you apply f to the range of g.
What is the difference between y = |f(x)| and y = f(|x|)?
For y = |f(x)|, you reflect any part of the graph below the x-axis upward, so the whole graph sits on or above the x-axis. For y = f(|x|), you keep the part of the graph for x ≥ 0 and reflect it across the y-axis, producing a graph symmetric about the y-axis. They look different and are a frequent source of lost marks, so sketch each carefully.
Do you need a graphing calculator for this topic?
Yes. H2 Mathematics assumes a graphing calculator throughout, and functions and graphs questions expect you to use it to confirm intercepts, asymptotes, stationary points and ranges. However, you must still produce a clear, labelled sketch and state results in correct notation — the calculator supports your working, it does not replace the method marks the examiner awards for showing reasoning.
Related: H2 Mathematics Overview · H2 Math Calculus · Sequences & Series · Differential Equations · Vectors, Lines & Planes