Relations and Functions

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Relations and Functions
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Learning Objectives

• Basic concepts.
• Function notation and the algebra of functions.
• Distance and slope formulas.

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Basic Concepts

• Cartesian product the cartesian product A X B is
  defined by
• and contains all ordered pairs where the
  first element is an element of A and the second
  of B.
• Example find the cartesian product of Ap,q,r
  and B3,5.
• A X B (p,3), (p,5), (q,3), (q,5), (r,3),
  (r,5)
• It can be represented by a table.

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Basic Concepts

• Relation a relation is any subset of the
  cartesian product. It is a set of ordered pairs.
• For example we can define a relation on R, where
  relation is a subset of R X R.
• Example the relation (1,1), (2,2), (3,2),
  (4,3) .
• domain 1,2,3,4, range
  1,2,3
• Example the relation (x,y) / x2 y2 1 .
  domain x/ -1 ? x ? 1, range y/
  -1 ? y ? 1

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Basic Concepts

• Example of relation
• the domain is A a, b, c, d
• the range is B Y, Z

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Basic Concepts

• Function a function is a set of ordered pairs or
  relation for which no two distinct ordered pairs
  have the same first element.
• It is also a mapping, rule or correspondence
  that assigns to each element x in set A exactly
  one element y in set B.
• Each x is mapped to exactly one y.
• f associates with each x in A one and only one y
  in B.
• A is called the domain and B is called the
  codomain.
• The range of f is the set of all images of points
  in A under f. We denote it by f(A).
• Vertical line test a graph is the graph of a
  function if no vertical line intersects the graph
  in more than one point.

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Basic Concepts

• Example the hyperbola graphed below is not a
  function.

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Basic Concepts

• Function examples (3,5), (4,9), (5,-7), (6,-7)
  is a function.
• (cubs, Chicago), (White Sox, Chicago),
  (Orioles, Baltimore), (Padres, San Diego) is a
  function.
• (Chicago, Cubs), (Chicago, White Sox),
  (Baltimore, Orioles), (San Diego, Padres) is
  not a function because Chicago is in two
  different pairs.

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Basic Concepts

• Function described with a set builder such as in
  (x,y) / y 3x 5. x is called the
  independent variable and y is called the
  dependent variable.
• The set of functions is a subset of the set of
  relations.
• Implicit domain the implicit domain is the set
  of all values for the independent variable that
  result in real number values for the dependent
  variable. We exclude values that would cause a
  division by zero or a square root of a negative
  number, for example.

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Basic Concepts

• Determine the domain of the function
• (x,y) / y
• We must exclude from the domain all values such
  as x2 4 0 , and x 5 lt 0 .
• Thus the domain -5, -2) ? (-2, 2) ? (2, ?).

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Function Notation

• We give a name to a function, such as f, g, h,
  etc.
• We represent the function f (x,y) / y 3x -
  5 by f(x) 3x-5.
• x is an independent variable, can be replaced by
  values, such as f(2) 1. Substitute 2 for each
  occurrence of x.

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Function Notation

• Definitions
• For all x common to the domains of f and g, the
  arithmetic of f and g is given by the
  followingthe sum is (fg)(x) f(x)
  g(x).the difference is (f-g)(x) f(x)
  g(x).the product is (fg)(x) f(x) g(x).the
  quotient is (f/g) (x) f(x) / g(x), for g(x) ? 0.

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Function Notation

• Example f(x) x2 x, g(x) x2. Find fg,
  f-g, fg, f/g.(fg)(x) f(x) g(x) x2 x
  x 2 x2 2.(f-g)(x) f(x) - g(x) x2 x
  - x - 2 x2 2x - 2.(fg)(x) f(x) g(x) (x2
  x) (x 2) x3 - x2 2x2 2x
  x3 x2 2x. (f/g)(x) f(x) / g(x) (x2 x)
  / (x 2), f/g is defined for x ? -2.

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Function Notation

• Definition for all x for which the expression
  is defined, the composition of functions,
  symbolized by (f o g)(x), is defined to be
  f(g(x)), sometimes written as fg(x) to avoid
  confusion between grouping symbols.(f o g) (x)
  is f at g(x).(f o g) exists for x where f is
  defined at g(x), which requires that g(x) is in
  the domain of f.

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Function Notation

• Examplef(x) (x-1) / x, g(x) (f o g) (x)
  fg(x) (g o f) (x) gf(x)

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Distance and Slope Formulas

• Pythagorean theorem the sum of the squares of
  the lengths of the legs in a right triangle is
  equal to the square of the length of the
  hypotenuse.
• leg12 leg22 hypotenuse2
• Distance formula the distance between two points
  (x1, y1) and (x2, y2) is

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Distance and Slope Formulas

• Example find the distance between the points (1,
  -2) and (5,4 ).

(5,4)
d
(1,-2)
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Distance and Slope Formulas

• Slope of a line a nonvertical line passing
  through the points (x1, y1) and (x2, y2) has a
  slope m

(x2, y2)
y2- y1
d
(x1, y1)
x2- x1
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Distance and Slope Formulas

• Example find the slope of the line of equation
  3x2y4.
• 1) -(3/2)x 2 (the slope is 3/2).
• 2) Take points (0, 2) and (4/3, 0).