Relations and Functions

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Relations and Functions
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Learning Objectives

• Variation.
• Inverse relations and functions.

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Variation

• Variation a variation is a function involving
  two or more variables.
• Direct variation y varies directly as, or is
  proportional to, x means y kx, where k ?
  0.Direct variation produces pairs of numbers in
  which the ratio (k) is constant.
• Example quantity varies directly as the square
  root of the radius y k ? x

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Variation

• Inverse variation y varies inversely as, or is
  inversely proportional to, x means y k/x, where
  k ? 0. k is called the constant of variation or
  the constant of proportionality. Example time
  varies inversely as the number of people who
  work t k / P.

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Variation

• Joint variation y varies jointly as x and z
  means y kxz, where k ? 0. k is called the
  constant of variation or the constant of
  proportionality. Example volume varies jointly
  as height and square of radius V k h r2.

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Variation
Direct variation
Inverse variation
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Inverse Relations and Functions

• Inverse function the inverse of a relation or
  function is the relation that results from
  interchanging the x- and y-coordinates in all
  ordered pairs.
• Example the inverse of the relation (1,2),
  (3,4) is the relation (2,1), (4,3) .The
  inverse of the function f(x) 2x1 is the
  relation g(x) (x-1) / 2.y 2x 1 ? 2x y-1
  ? x (y-1) / 2.

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Inverse Relations and Functions

• The inverse function is noted f-1.
• The graph of the inverse function is the mirror
  image of the function f in the line y x.

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Inverse Relations and Functions

• The inverse function is obtained in Maple by
  solve
• solve( y 2x 1, x) ? x ½ y ½

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Inverse Relations and Functions

• Properties of the inverseff-1(x) f-1f(x)
  xfo f-1 f-1of Id (the identity function
  Id(x) x).
• To find the inverse of f(x), write y f(x), and
  solve for x, to get x f-1(y). Then replace y by
  x to obtain f-1(x).
• Example f(x) x3 2, find f-1.y x3 2y-2
  x3 x 3? y-2and
• f-1(x) 3? x-2.

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Inverse Relations and Functions

• The inverse of a function is not always a
  function.
• Example f(x) x2 3, f-1(x) ?? x-3.Most
  xs will have two images, thus f-1(x) is not a
  function.
• A one-to-one function is a function for which
  each element in the codomain is the image of a
  single x.
• The inverse of a one-to-one function is a
  function.
• Example a line is a one-to-one function.