Functions

Functions describe how one quantity depends on another. Learn notation, domain, codomain and range, independent and dependent variables, piecewise, composite and inverse functions, even/odd and periodic functions, monotonicity and boundedness, and the families of algebraic, trigonometric, exponential and logarithmic functions used to model temperature(t), position(t), voltage(t) and pressure(x).

A function is a rule assigning exactly one output to each input. Engineers use functions as models: the voltage across a capacitor as a function of time, the stress in a beam as a function of position, fuel consumption as a function of speed.

An analogy

A function is like a vending machine: each button (input) always gives one specific item (output). Two buttons can give the same item, but one button never gives two items.

Domain and range

Domain excludes values that break the formula: division by zero, square roots of negatives (in ℝ), logs of non-positive numbers — and physically impossible inputs (negative time, negative mass).

Building functions

Piecewise functions model switches and tariffs (e.g. tax brackets, a thermostat). Composition chains processes: sensor voltage → temperature → alarm. Inverses undo them — only one-to-one functions have inverses (restrict the domain otherwise).

Symmetry and periodicity

cos is even, sin is odd; every function splits into even + odd parts. Periodic functions (AC voltage, rotating shafts) repeat every T; frequency f = 1/T.

Function families

Polynomials (smooth, unbounded), rational functions (asymptotes), exponentials/logs (growth, decay), trigonometric (oscillations), power laws (scaling). Choosing the right family is the first step of mathematical modelling.

Formulas

• Composition: (f∘g)(x) = f(g(x))
• Inverse: f(f⁻¹(x)) = x — swap x and y, solve for y
• Even / odd: Even: f(−x) = f(x); Odd: f(−x) = −f(x)
• Periodic: f(x + T) = f(x) for all x

Worked example: Inverse function

Given: y = (2x + 3)/(x − 1)

Find: f⁻¹ and domains.

• Swap: x = (2y + 3)/(y − 1)
• xy − x = 2y + 3 → y(x − 2) = x + 3
• f⁻¹(x) = (x + 3)/(x − 2)

Answer: Domain of f: x ≠ 1; domain of f⁻¹: x ≠ 2

Worked example: Piecewise tariff

Given: Cost = 0.10 $/kWh up to 100 kWh, 0.15 $/kWh above

Find: Cost for 160 kWh.

• 100 × 0.10 = $10
• 60 × 0.15 = $9

Answer: $19

Common mistakes

• Wrong: Ignoring domain restrictions. — Right: State where the function is defined (and physically meaningful).
• Wrong: Confusing f⁻¹(x) with 1/f(x). — Right: Inverse undoes f; reciprocal is 1/f.

Applications

• Sensor calibration: Inverse functions convert readings back to physical values.
• Control logic: Piecewise functions for saturation and thresholds.
• Signals: Periodic functions model AC and vibrations.

Key takeaways

• One input → one output.
• State domain and range.
• Compose to chain, invert to undo.
• Even, odd, periodic symmetries.

Going further

• Multivariable functions: f(x, y) — see Module M3.
• Functional equations: Finding f from its properties.
• Function spaces: Functions as vectors (functional analysis).