H2 Maths Notes (JC 1-2): 1.1) Functions

Before you revise\ Keep a running list of domain restrictions (denominators, even roots, logarithms). Test every claim with quick substitutions or GC trace—functions questions are marked on precision.

Core Concepts

• A function \( f: A \to B \) assigns each \( x \in A \) to one output \( f(x) \in B \); \( A \) is the domain, \( f(A) \) is the range.
• Injective (one-to-one) functions satisfy \( f(x_1) = f(x_2) \Rightarrow x_1 = x_2 \) and pass the horizontal-line test.
• Surjective (onto) functions cover the whole codomain; bijective functions are both injective and surjective and therefore invertible.
• Composition \( (f \circ g)(x) = f\bigl(g(x)\bigr) \) is only defined when \( g(x) \) lies inside the domain of \( f \).
• Inverses reverse the mapping: \( f^{-1}\bigl(f(x)\bigr) = x \) for all \( x \) in the restricted domain.

Domain and Range Workflow

1. Determine the natural domain by eliminating forbidden inputs:
   • Even roots: require the radicand \( \geq 0 \).
   • Logarithms: argument strictly \( > 0 \).
   • Rational functions: denominator \( \neq 0 \).
   • Composite functions: propagate restrictions through every stage.
2. State the range by analysing turning points, asymptotes, or monotonic intervals.
3. Restrict the domain if necessary to make \( f \) one-to-one before finding the inverse.

Example -- Square root domain

For \( f(x) = \sqrt{3x - 5} \), require \( 3x - 5 \geq 0 \) so \( x \geq \tfrac{5}{3} \). The range is \( [0, \infty) \).

Inverse Functions

• Use algebraic manipulation to isolate \( x \) in \( y = f(x) \); then interchange \( x \) and \( y \) to express \( f^{-1}(x) \).
• Confirm \( f^{-1}\bigl(f(x)\bigr) = x \) on the chosen domain and state both domain and range for the inverse explicitly.
• For quadratics, complete the square to reveal monotonic branches.

Example -- Quadratic inverse

Consider \( f(x) = 2x^2 - 3x - 1 \).

1. Stationary point from \( f'(x) = 4x - 3 = 0 \) gives \( x = \tfrac{3}{4} \).
2. Restrict domain to \( x \geq \tfrac{3}{4} \) so \( f \) is strictly increasing.
3. Complete the square: \( y = 2\bigl(x - \tfrac{3}{4}\bigr)^2 - \tfrac{25}{8} \).
4. Swap variables: \( x = 2\bigl(y - \tfrac{3}{4}\bigr)^2 - \tfrac{25}{8} \) (treating \( y \) as input) then solve for \( y \: \) \( f^{-1}(x) = \tfrac{3}{4} + \sqrt{ \frac{x + \tfrac{25}{8}}{2} } \).
5. The inverse domain is \( x \geq -\tfrac{25}{8} \) and range \( y \geq \tfrac{3}{4} \).

Composite Functions

• Record intermediate ranges meticulously: if \( g: A \to B \) and \( f: C \to D \), composition requires \( g(A) \subseteq C \).
• When solving \( (f \circ g)(x) = k \), set \( g(x) \) equal to an auxiliary variable first, solve \( f(u) = k \), then back-substitute.
• Use the identity \( (f \circ g)^{-1} = g^{-1} \circ f^{-1} \) only after verifying both inverses exist on the relevant domains.

Example -- Composition with logarithms

Let \( f(x) = \ln(x - 1) \) and \( g(x) = x^2 + 4 \).

1. Natural domain of \( g \) is all reals, range \( [4, \infty) \).
2. \( f \) requires input \( > 1 \), so \( g(x) > 1 \) holds for all \( x \).
3. Composition \( (f \circ g)(x) = \ln(x^2 + 3) \).
4. Inverse sequence: \( f^{-1}(x) = 1 + e^x \), \( g^{-1}(x) = \pm \sqrt{x - 4} \) (choose the positive branch if restricting to \( x \geq 0 \)).

Modelling with Functions

• When modelling real data, specify domain in context (e.g. weeks, temperatures).
• Piecewise definitions handle regime changes: \[ f(x) = \begin{cases} 2x + 3, & x < 1 \\ x^2, & x \geq 1 \end{cases} \]
• Always state continuity and differentiability at junctions.

Calculator Workflow

• Use TABLE mode to scan domain issues quickly (undefined entries highlight restrictions).
• CALC > min / max helps verify monotonic intervals before restricting domains.
• Document the command sequence in worked solutions, e.g. “GC: ISCT between \( y_1 = 2x^2 - 3x - 1 \) and \( y_2 = k \).”

Quick Revision Checklist

• [ ] Identify domains and ranges, quoting restrictions explicitly.
• [ ] Prove injectivity via algebra or derivative sign to justify inverses.
• [ ] Execute compositions and inverses while checking compatibility of domains.
• [ ] Translate contextual descriptions into precise function definitions.

Next steps: Move on to 1.2 for a deeper dive into graph transformations paired with these functional foundations.