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mathematics

Function approximation

Function approximation is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Function approximation rather than just read about it. In short: In general, a function approximation problem asks us to select a function that closely matches ("approximates") a function in a task-specific way. The need for function approximations arises, for example, predicting the growth of microbes in microbiology.

Function approximation — main illustration
Function approximation — illustration

Key takeaways

  • Function approximation belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Function approximation to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Function approximation from memory before moving on to harder problems.

Reference excerpt

In general, a function approximation problem asks us to select a function that closely matches ("approximates") a function in a task-specific way. The need for function approximations arises, for example, predicting the growth of microbes in microbiology. Function approximations are used where theoretical models are unavailable or hard to compute. First, for known target functions approximation theory is the branch of numerical analysis that investigates how certain known functions (for example, special functions) can be approximated by a specific class of functions (for example, polynomials or rational functions) that often have desirable properties (inexpensive computation, continuity, integral and limit values, etc.). Secondly, for example, if g is an operation on the real numbers, techniques of interpolation, extrapolation, regression analysis, and curve fitting can be used. If the codomain (range or target set) of g is a finite set, one is dealing with a classification problem instead.

See also Approximation theory Fitness approximation Kriging Least squares (function approximation) Radial basis function network

References

Illustrations

Function approximation: Several progressively more accurate approximations of the step function
Several progressively more accurate approximations of the step function
Function approximation: An asymmetrical Gaussian function fit to a noisy curve using regression
An asymmetrical Gaussian function fit to a noisy curve using regression

Worked examples

Example 1 — a first encounter with Function approximation

Start with the simplest possible case. Write down what Function approximation claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Function approximation before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Function approximation ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Function approximation

In research
Function approximation appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Function approximation in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Function approximation is common in secondary-school and first-year university syllabi. It links to neighbouring topics Mathematical analysis stubs, Regression analysis, Statistical approximations, so understanding it makes those chapters shorter.
In everyday life
Look for Function approximation outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.
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How to study Function approximation in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Function approximation means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Function approximation out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Function approximation in simple terms?

In general, a function approximation problem asks us to select a function that closely matches ("approximates") a function in a task-specific way. The need for function approximations arises, for example, predicting the growth of microbes in microbiology.

Why does Function approximation matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Function approximation?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Function approximation.

Tags

  • Mathematical analysis stubs
  • Regression analysis
  • Statistical approximations
  • Statistics stubs

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