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RnF - Coggle Diagram

- - - - Empty
      - Universal
    - - Symmetric
      - Transitive
      - Reflexive
  - - - How to know if a relation is a function
        
        Algebraically
        
        Independent and dependent variables
        
        Graphically
        
        Vertical line test
      - Terms
        
        Codomain = set B
        
        Range
        
        Domain = Set A
      - Algebra of function
        
        Substract
        
        Multiply
        
        Add
        
        Divide
    - - Constant inside function
      - Constant outside function
      - Modulus
      - GIF
      - operations
        
        Stretch
        
        Shift
        
        Squeeze
        
        Flip
        
        Horizontal
        
        Vertical
    - - Transcedental
        
        Exponential
        
        f(x) = a^x where a>1 and x e R
        
        TF
        
        tanx
        
        cosecx
        
        range of asinx + bcosx
        
        secx
        
        cosx
        
        cotx
        
        sinx
        
        Inverse functions
        
        Log (inverse)
        
        ITF(Inverse)
        
        Periodic fun
      - Algebraic
        
        Piecewise function
        
        Signum
        
        Fractional
        
        GIF
        
        Modulus
        
        Function of the form f(x) max or min
        
        Polynomial
        
        Quadratic
        
        vertex : (-b/2a, -D/4a)
        
        Case 2: a<0
        
        Case 1 : a>0
        
        If odd then has range R
        
        If even then does not take all R values
        
        Rational
        
        f(x) = P(x)/Q(x)
      - Some basic/ Elementary functions
        
        Constant fun
        
        Square fun
        
        Cubic fun
        
        Reciprocal fun
        
        Identity fun
        
        Identical functions
        
        Condition for two fucntions to be identical
        
        Modulus fun
      - Based on
        
        Symmetry of curve
        
        Evevn and odd
        
        Even and odd extension
        
        Mappings
        
        One-one and many-one
        
        One - one
        
        Has distinct image in codomain
        
        Also called injective
        
        Many-one
        
        Two pre images with same image
        
        Methods to determine one one and many one
        
        Graphical
        
        at least on eline parallel to x-axis cutting graph in two points then many one else one one
        
        Differentiation
        
        For continuous function f`(x) > or < 0 for all x belongs to domain of fun then it is one one else many one
        
        Analytical
        
        f(x1) =/= f(x2) -> one one else many one
        
        Onto- Into
        
        Onto
        
        Also known as surjective
        
        Range of f is same as codomain
        
        Into
        
        Range is subset of codomain
        
        Methods to determine onto and into fun
        
        Independent -> dependent
        
        Explicit and Implicit
        
        Has boundded or not
        
        Bounded
        
        Unbounded
      - Others
        
        Composite
    - - Finding function
      - Finding value of function at some point
      - Some properties
  - - - Range