A precise definition
A function is a rule that assigns exactly one output to each input. The notation f(x) means: "apply rule f to input x." That simple idea — one input, exactly one output — turns out to be the language in which every scientific law, economic model, and engineering formula is written. Demand as a function of price. Temperature as a function of time. Population as a function of resources. The function concept unifies all of these into one framework.
The problem it was invented to solve
The formalism of functions was established by Euler in the 18th century, but the idea existed wherever mathematicians described relationships. The reason it became central is that it provides a precise language: when you say 'y = f(x)', you are making an exact, unambiguous claim about how y depends on x — not a vague description, but a complete specification. Science runs on this.
Where you find it in the world — including South Africa
These are not contrived textbook examples. Each application below is currently in use, driven by real institutions, and producing real outcomes.
South African GDP as a function of time
Stats SA publishes GDP data as a function of time. Economists fit mathematical functions to this data, then differentiate those functions to find the growth rate and integrate them to find cumulative output. Every economic forecast is function analysis.
Eskom load curves: power consumption over 24 hours
The daily load curve — electricity demand plotted against time — is a function. Eskom engineers analyse this function to predict peaks, model generation requirements, and schedule maintenance. Load shedding decisions are made by analysing where this function exceeds generation capacity.
Population modelling and demographic projections
South Africa's population projections (published by Stats SA) use exponential and logistic functions to model growth under different fertility and mortality scenarios. These projections drive government planning for schools, hospitals, and housing.
Drug dosage and pharmacokinetics
When a doctor prescribes a drug 'twice daily', that interval is calculated from the drug's half-life — which is modelled as an exponential decay function. The function f(t) = A·e^(-kt) describes concentration in the bloodstream, and the prescription is set so concentrations stay in the therapeutic range.
Interest rates and banking
The growth of an investment or a debt is an exponential function of time. Every bond repayment schedule at FNB, ABSA, or Standard Bank is a function evaluated at monthly intervals. The difference between linear growth and exponential growth — which every South African borrower should understand — is the difference between f(x)=mx and f(x)=aˣ.
You've already encountered this
Every time you check your data remaining or your account balance online, the platform is evaluating functions — converting time elapsed and rate of use into a single output. The 'recommended for you' algorithm on streaming services is a function that maps your viewing history to a predicted rating for each unwatched title.
What you study — and when
- ›Linear functions: gradient, y-intercept, parallel and perpendicular lines
- ›Quadratic functions: parabolas, vertex form, roots, discriminant
- ›Hyperbolic functions: asymptotes, transformations
- ›Exponential functions: growth and decay, graphs, applications
- ›Trigonometric functions: sin, cos, tan graphs and transformations
- ›Logarithmic functions (Gr12): relationship to exponentials, equations
Related topics and institutions
Functions are the thread connecting every topic in your CAPS curriculum.
The Continuum helps you build a complete, connected understanding of the function family — so that when you encounter a new type of function in Grade 11 or 12, it feels like meeting a relative, not a stranger.
No card required. South African curriculum. Grade 8–12.