Math 2.0

definition

Rational function

Definition

Let pp and qq be real polynomials, with qq not the zero polynomial. Let D={x∈R:q(x)≠0}D = \{ x \in \mathbb{R} : q(x) \ne 0 \}. The function f ⁣:D→Rf \colon D \to \mathbb{R} given by f(x)=p(x)/q(x)f(x) = p(x)/q(x) is a rational function.

Remarks

A nonzero polynomial has only finitely many roots, so DD is R\mathbb{R} minus a finite set. Each rational function is continuous on DD, since polynomials are continuous and quotients of continuous functions are continuous. For example, 1/x1/x and (x2+1)/(x−2)(x^2 + 1)/(x - 2) are rational functions.

Explanations

L1 human

A rational function is a quotient of two polynomials. It is defined everywhere except at the zeros of the denominator.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionReal sequence, definition, no evidence levelReal sequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionEpsilon-delta definition of continuity, statement, no evidence levelEpsilon-delta definit…statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionEpsilon-N definition of the limit of a sequence, statement, no evidence levelEpsilon-N definition …statementSequential continuity, definition, no evidence levelSequential continuitydefinitionContinuity is equivalent to sequential continuity, statement, no evidence levelContinuity is equival…statementConvergent sequences are bounded, statement, no evidence levelConvergent sequences …statementLimit of a product of sequences, statement, no evidence levelLimit of a product of…statementProduct of continuous functions is continuous, statement, no evidence levelProduct of continuous…statementLimit of a quotient of sequences, statement, no evidence levelLimit of a quotient o…statementPolynomial functions are continuous, statement, no evidence levelPolynomial functions …statementQuotient of continuous functions is continuous, statement, no evidence levelQuotient of continuou…statementRational function, definition, no evidence levelRational functiondefinition
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  1. Limit point of a set definition
  2. Sequence definition
  3. Limit of a function definition
  4. Real sequence definition
  5. Continuity at a point definition
  6. Bounded sequence definition
  7. Limit of a sequence definition
  8. Continuous function definition
  9. Epsilon-delta definition of continuity statement
  10. Convergent sequence definition
  11. Epsilon-N definition of the limit of a sequence statement
  12. Sequential continuity definition
  13. Continuity is equivalent to sequential continuity statement
  14. Convergent sequences are bounded statement
  15. Limit of a product of sequences statement
  16. Product of continuous functions is continuous statement
  17. Limit of a quotient of sequences statement
  18. Polynomial functions are continuous statement
  19. Quotient of continuous functions is continuous statement
  20. Rational function definition

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Sources

  1. Web https://en.wikipedia.org/wiki/Rational_function, Definitions

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