function and their proporties

这是一篇关于function and their proporties的思维导图，主要内容有Definition of a function、Domain、Range等。

function and their proporties

• Definition of a function
  • The function must contains the equation and it must have unknowns
  • A function from a set D(domain) to another set R(range) is a rule that assighs to every element in D a unique element in R.
• Domain
  • Always represented by x (is the independent variable)
  • All input of a function
  • We match domain values along the x-axis
• Range
  • Always represented by y(is the dependent variable) and we also use f(x) to indicate that y comes from the function acting on x
  • All output values of a function
  • Uniqueness is vital for range and we match the range along the y-axis
• Properties of a function
  • continuity
    • continuuous function: if the function doesn't come apart at any point
    • Removable discontinuity
      • The graph is continuous everywhere except for a "hole"
      • It can be patched by redefining f(x) so as to plug the hole
    • Non-removable
      • Jump discontinuity
        • There is a jump in function values that make the gap impossible to plug with a single point
      • Infinite dis continuity
        • This is a function with an infinite discontinuity
  • Change in function
    • Increasing
      • Positive change in x results in a positive change in f(x)
    • Dcreasing
      • Positive change in x results in negative change in f(x)
    • Constant
      • Positive change in x results in a no change in f(x)
  • Boundedness
    • Bounded above
      • If there is some number B that is greater than or equal to every number in the range of the function.
    • Bounded below
      • If there is some number B that is less than or equal to every number in the range of the function.
    • Bounded
      • If the function is bounded that means it is bounded both above and below .
    • Not bounded
      • If the function is not bounded that means it is neither bounded above nor below.
  • Local and Absolute extrema
    • Maximum
      • local maximum
        • Always characterized by peaks
        • It is a value that is greater than or equal to all range values of the function ( on some open interval of this graph)
      • Absolute maximum
        • It is a value that is greater then or equal to all range value of this function
    • Minimum
      • Local minimum
        • It is a value that is less than or equal to all range values of the function (in some open interval of this function)
      • Absolute minimum
        • It is a value that is less than or equal to all range values of this function
  • Symmetry
    • Reflection symmetry
      • Even function
        • f(-x)=f(x)
    • Rotational symmetry
      • Odd function
        • f(-x)=-f(x)
  • Asymptotes
    • Horizontal asymptotes
      • The limit of the function when x approaches negative infinity equals b
    • Vertical asymptotes
      • The limit of the function when x approaches a from left or right equals positive or negative infinity.
  • End behavior
    • End behavior shows how the function behave as it goes toward either "end" of x-axis. A horizontal asymptotes gives one kind of end behavior.