H2 Maths Notes (JC 1-2): 1) Functions and Graphs

Before you begin\ Make sure your Additional Maths foundations are fresh (logarithms, surds, quadratic inequalities). Keep the 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 \( (f \circ g)(x) = f\bigl(g(x)\bigr) \) 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 \( x = \tfrac{3}{4} \).
2. On the interval \( [\tfrac{3}{4}, \infty) \) the derivative is non-negative, so the function is one-to-one.
3. Completing the square gives \( f(x) = 2\bigl(x - \tfrac{3}{4}\bigr)^2 + \tfrac{23}{8} \); the inverse is \( f^{-1}(y) = \tfrac{3}{4} + \sqrt{ \frac{y - \tfrac{23}{8}}{2} } \).

Composition pitfalls

• Always record the intermediate range when composing functions.
• Do not rely on the shortcut \( (f \circ g)^{-1} = g^{-1} \circ f^{-1} \); 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 \( \frac{ax + b}{cx + d} \) 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

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 \( y = -2 f\bigl( \tfrac{x}{3} \bigr) + 1 \) 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. \( \lvert x - a \rvert < b \) implies \( a - b < x < a + b \).

Example 3 -- Rational inequality

Solve \( \frac{x - 2}{x + 1} > 0 \).

1. Critical points: \( x = -1 \) (vertical asymptote) and \( x = 2 \) (zero).
2. Test intervals \( (-\infty, -1) \), \( (-1, 2) \), \( (2, \infty) \).
3. The inequality holds on \( (-1, 2) \) and \( (2, \infty) \); exclude \( x = -1 \).

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.

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).