Unit 3 - Objective 1 - Dependent and Independent Variables


A relation is used to represent a pairing (relationship) between two numbers, variables, or objects.  A function is a special kind of relation.  For any relation, the set of all possible first component values is called the domain and the set of all second values that can result from using values in the domain is called the range.   If a relation between x and y has only one value of y for each value of x, then we say that y is a function of x.  Every function is a relation but not every relation is a function.

Function
The variable y is a function of x if a relation between x and y produces exactly one value of y for each value of x.  The first variable is called the independent variable and the second variable is called the dependent variable.   The phrase "function of x" is written as f(x).

Examples:

  1. For a function y = 7x - 2, the variable x is called the independent variable and y is the dependent variable.  The value y depends upon the value selected for x.  The number that corresponds to y is designated as f(x), g(x), h(x) depending on how the function is identified.
  2. Suppose you have the function f(x) = 2x2 - 3 and you want to know what value of f which corresponds to x = 4.  You are asking for f(4) and you substitute 4 for each x in the function.  The result is:
  3. f(4) = 2 (4)2 - 3
    = 2 (16) - 3
    = 32- 3
    = 29

    The value of f(4) is 29.  This gives the ordered pair (4, f(4)) or (4,29)

     

  4. If f(x) = 3x2 - 5x + 2, find f(- 2)
  5. f(- 2) = 3 (- 2)2 - 5 (-2) + 2
    = 3 (4) + 10 + 2
    = 12 + 10 + 2
    = 24

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