Rational function

Key takeaways

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

Reference excerpt

In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field K. In this case, one speaks of a rational function and a rational fraction over K. The values of the variables may be taken in any field L containing K. Then the domain of the function is the set of the values of the variables for which the denominator is not zero, and the codomain is L. The set of rational functions over a field K is a field, the field of fractions of the ring of the polynomial functions over K.

Definitions A function f {\displaystyle f} is called a rational function if it can be written in the form

f ( x ) = P ( x ) Q ( x ) {\displaystyle f(x)={\frac {P(x)}{Q(x)}}}

where P {\displaystyle P} and Q {\displaystyle Q} are polynomial functions of x {\displaystyle x} and Q {\displaystyle Q} is not the zero function. The domain of f {\displaystyle f} is the set of all values of x {\displaystyle x} for which the denominator Q ( x ) {\displaystyle Q(x)} is not zero. However, if P {\displaystyle \textstyle P} and Q {\displaystyle \textstyle Q} have a non-constant polynomial greatest common divisor R {\displaystyle \textstyle R} , then setting P = P 1 R {\displaystyle \textstyle P=P_{1}R} and Q = Q 1 R {\displaystyle \textstyle Q=Q_{1}R} produces a rational function

f 1 ( x ) = P 1 ( x ) Q 1 ( x ) , {\displaystyle f_{1}(x)={\frac {P_{1}(x)}{Q_{1}(x)}},}

which may have a larger domain than f {\displaystyle f} , and is equal to f {\displaystyle f} on the domain of f . {\displaystyle f.} It is a common usage to identify f {\displaystyle f} and f 1 {\displaystyle f_{1}} , that is to extend "by continuity" the domain of f {\displaystyle f} to that of f 1 . {\displaystyle f_{1}.} Indeed, one can define a rational fraction as an equivalence class of fractions of polynomials, where two fractions A ( x ) B ( x ) {\displaystyle \textstyle {\frac {A(x)}{B(x)}}} and C ( x ) D ( x ) {\displaystyle \textstyle {\frac {C(x)}{D(x)}}} are considered equivalent if A ( x ) D ( x ) = B ( x ) C ( x ) {\displaystyle A(x)D(x)=B(x)C(x)} . In this case P ( x ) Q ( x ) {\displaystyle \textstyle {\frac {P(x)}{Q(x)}}} is equivalent to P 1 ( x ) Q 1 ( x ) . {\displaystyle \textstyle {\frac {P_{1}(x)}{Q_{1}(x)}}.}

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Rational function

Start with the simplest possible case. Write down what Rational function 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 Rational function 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 Rational function 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 Rational function

In research: Rational function 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 Rational function 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: Rational function is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic varieties, Meromorphic functions, Morphisms of schemes, so understanding it makes those chapters shorter.
In everyday life: Look for Rational function 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.

How to study Rational function in 20 minutes

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

Frequently asked questions

What is Rational function in simple terms?

In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field K.

Why does Rational function 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 Rational function?

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 Rational function.