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.
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∘g)(x)=f(g(x)) requires the range of g
to sit within the domain of
f
.
Domain/range checks
State the natural domain (avoid zero denominators, even roots of negatives, logarithms of non-positive numbers).
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)=2x2−3x+4.
f′(x)=4x−3 has a stationary point at x=43.
On the interval [43,∞) the derivative is non-negative, so the function is one-to-one.
Completing the square gives f(x)=2(x−43)2+823
Composition pitfalls
Always record the intermediate range when composing functions.
Do not rely on the shortcut (f∘g)−1=g−1∘f−1; verify manually.
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; classify them using the sign of f′ or the second derivative.
For rational functions cx+dax+b determine vertical asymptotes from the denominator and horizontal/oblique asymptotes via division.
Check symmetry: even f(−x)=f(x) and odd f(−x)=−f(x).
Transformation cheat sheet
Transformation
Effect on y=f(x)
Notes
y=af(x)
Vertical stretch by ∣a∣ (reflect if a<0)
Multiply y-values.
y=f(x)+a
Shift up by a
Add to the output.
y=f(x+a)
Shift left by a
Replace x with x+a
y=f(ax)
Horizontal compression by ∣a∣ (reflect if a<0)
Divide x-values by a
y=∣f(x)∣
Reflect negative parts above the x-axis
Ensure the range is non-negative.
y=f(∣x∣)
Mirror the right-half graph across the y-axis
Only the right half is independent.
y=f(x)1
Reciprocal graph
Watch for new vertical asymptotes where f(x)=0
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) and h(t).
Example 2 -- Transformation workflow
Sketch y=−2f(3x)+1 given y=f(x).
Start with the original graph.
Stretch horizontally by factor 3 (replace x with 3x).
Stretch vertically by factor 2 and reflect across the x-axis (multiply by −2).
Translate the result up by 1 unit.
1.3 | Equations and Inequalities
General equation strategy
Formulate the equation from the context.
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. ∣x−a∣<b implies a−b<x<a+b.
Example 3 -- Rational inequality
Solve x+1x−2>0.
Critical points: x=−1 (vertical asymptote) and x=2 (zero).
Test intervals (−∞,−1), (−1,2), (2,∞).
The inequality holds on (−1,2) and (2,∞); exclude x=−1.
Example 4 -- Absolute inequality
Solve ∣2x−3∣≤5.
Convert to −5≤2x−3≤5.
Hence −1≤x≤4.
Graphical method with GC
Plot y=f(x).
Use the intersection tool to locate roots or boundary points.
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).
