one-valued function

Function \Func"tion\, n. [L. functio, fr. fungi to perform,
   execute, akin to Skr. bhuj to enjoy, have the use of: cf. F.
   fonction. Cf. {Defunct}.]
   1. The act of executing or performing any duty, office, or
      calling; performance. "In the function of his public
      calling." --Swift.
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   2. (Physiol.) The appropriate action of any special organ or
      part of an animal or vegetable organism; as, the function
      of the heart or the limbs; the function of leaves, sap,
      roots, etc.; life is the sum of the functions of the
      various organs and parts of the body.
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   3. The natural or assigned action of any power or faculty, as
      of the soul, or of the intellect; the exertion of an
      energy of some determinate kind.
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            As the mind opens, and its functions spread. --Pope.
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   4. The course of action which peculiarly pertains to any
      public officer in church or state; the activity
      appropriate to any business or profession.
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            Tradesmen . . . going about their functions. --Shak.
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            The malady which made him incapable of performing
            his
            regal functions.                      --Macaulay.
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   5. (Math.) A quantity so connected with another quantity,
      that if any alteration be made in the latter there will be
      a consequent alteration in the former. Each quantity is
      said to be a function of the other. Thus, the
      circumference of a circle is a function of the diameter.
      If x be a symbol to which different numerical values can
      be assigned, such expressions as x^{2}, 3^{x}, Log. x, and
      Sin. x, are all functions of x.
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   6. (Eccl.) A religious ceremony, esp. one particularly
      impressive and elaborate.

            Every solemn `function' performed with the
            requirements of the liturgy.          --Card.
                                                  Wiseman.
      [Webster 1913 Suppl.]

   7. A public or social ceremony or gathering; a festivity or
      entertainment, esp. one somewhat formal.

            This function, which is our chief social event. --W.
                                                  D. Howells.
      [Webster 1913 Suppl.]

   {Algebraic function}, a quantity whose connection with the
      variable is expressed by an equation that involves only
      the algebraic operations of addition, subtraction,
      multiplication, division, raising to a given power, and
      extracting a given root; -- opposed to transcendental
      function.

   {Arbitrary function}. See under {Arbitrary}.

   {Calculus of functions}. See under {Calculus}.

   {Carnot's function} (Thermo-dynamics), a relation between the
      amount of heat given off by a source of heat, and the work
      which can be done by it. It is approximately equal to the
      mechanical equivalent of the thermal unit divided by the
      number expressing the temperature in degrees of the air
      thermometer, reckoned from its zero of expansion.

   {Circular functions}. See {Inverse trigonometrical functions}
      (below). -- Continuous function, a quantity that has no
      interruption in the continuity of its real values, as the
      variable changes between any specified limits.

   {Discontinuous function}. See under {Discontinuous}.

   {Elliptic functions}, a large and important class of
      functions, so called because one of the forms expresses
      the relation of the arc of an ellipse to the straight
      lines connected therewith.

   {Explicit function}, a quantity directly expressed in terms
      of the independently varying quantity; thus, in the
      equations y = 6x^{2}, y = 10 -x^{3}, the quantity y is an
      explicit function of x.

   {Implicit function}, a quantity whose relation to the
      variable is expressed indirectly by an equation; thus, y
      in the equation x^{2} + y^{2} = 100 is an implicit
      function of x.

   {Inverse trigonometrical functions}, or {Circular functions},
      the lengths of arcs relative to the sines, tangents, etc.
      Thus, AB is the arc whose sine is BD, and (if the length
      of BD is x) is written sin ^{-1}x, and so of the other
      lines. See {Trigonometrical function} (below). Other
      transcendental functions are the exponential functions,
      the elliptic functions, the gamma functions, the theta
      functions, etc.

   {One-valued function}, a quantity that has one, and only one,
      value for each value of the variable. -- {Transcendental
   functions}, a quantity whose connection with the variable
      cannot be expressed by algebraic operations; thus, y in
      the equation y = 10^{x} is a transcendental function of x.
      See {Algebraic function} (above). -- {Trigonometrical
   function}, a quantity whose relation to the variable is the
      same as that of a certain straight line drawn in a circle
      whose radius is unity, to the length of a corresponding
      are of the circle. Let AB be an arc in a circle, whose
      radius OA is unity let AC be a quadrant, and let OC, DB,
      and AF be drawnpependicular to OA, and EB and CG parallel
      to OA, and let OB be produced to G and F. E Then BD is the
      sine of the arc AB; OD or EB is the cosine, AF is the
      tangent, CG is the cotangent, OF is the secant OG is the
      cosecant, AD is the versed sine, and CE is the coversed
      sine of the are AB. If the length of AB be represented by
      x (OA being unity) then the lengths of Functions. these
      lines (OA being unity) are the trigonometrical functions
      of x, and are written sin x, cos x, tan x (or tang x), cot
      x, sec x, cosec x, versin x, coversin x. These quantities
      are also considered as functions of the angle BOA.
      Function