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Functions - Coggle Diagram

- - - - We say that the points connect the two variables.
      - How we write it?
        
        A relation is often expressed in the form of an equation connecting the variables x and y.
      - an Example for it
        
        y= x+3
        
        They are the equations of two relations.
    - - Not all relations can be defined by equation.
        
        Examples
        
        1) Set of all points in the first quadrant is the relation x> 0, x> 0
        
        2) These 13 points from a relation. It can be described as a finite set of points, but not by an equation
        
        They are not consistent enough to put them in a equation.
  - - - A Function is a relation in which no two different ordered pairs have the same x-coordinate or first component.
    - - We can see from this definition that a function is a special type of relation.
      - Every function is a relation, but not every relation is a function.
  - - - Suppose a relation is given as an equation. If the substitution of any value for results in at most one value of y, then the relation is a function.
    - - y = 3x -1 is a function , since for any value of x there is only one corresponding value of y.
  - - - Suppose we draw all possible vertical lines on the graph of a reflection
      - The Test Rules:
        
        1-) If each line cuts the graph at most once, then the relation is a function
        
        2-) If at least one line cuts the graph more than once , then the relation is not a function
        
        Let's Show it:
- - - - is a group of functions with graphs that display one or more similar characteristics.
    - - is the simplest function with the defining characteristics of the family. Functions in the same family are transformations of their parent functions.
  - - - Rule
        
        𝒇(𝒙)=𝒄
      - Range
        
        [𝒄]
      - Domain
        
        (−∞,∞)
      - Graph
    - - Domain
        
        (−∞,∞)
      - Range
        
        (−∞,∞)
      - Rule
        
        𝒇(𝒙)=𝒙
      - Graph
    - - Rule
        
        𝒇(𝒙)=𝒙^𝟐
      - Range
        
        [𝟎,∞)
      - Graph
      - Domain
        
        (−∞,∞)
    - - Domain
        
        (−∞,∞)
      - Range
        
        (−∞,∞)
      - Rule
        
        𝒇(𝒙)=𝒙^𝟑
      - Graph
    - - Domain
        
        [𝟎,∞)
      - Range
        
        =[𝟎,∞)
      - Rule
        
        𝒇(𝒙)=√𝒙
      - Graph
    - - Rule
        
        𝒇(𝒙)=𝟏/𝒙
      - 𝑫
        
        (−∞,𝟎)∪(𝟎,∞)
      - 𝑹
        
        (−∞,𝟎)∪(𝟎,∞)
      - Graph
    - - Rule
        
        𝒇(𝒙)=|𝒙|
        
        from that we can also say
        
        |𝒙|=
        
        −𝒙 𝒊𝒇 𝒙<𝟎
        
        𝒙 𝒊𝒇 𝒙≥0
      - 𝑫
        
        (−∞,∞)
      - 𝑹
        
        [𝟎,∞)
      - Graph
  - - - Rigid transformations
        
        change only the position of the graph, leaving the size and shape unchanged.
      - Non rigid transformations
        
        distort the shape of the graph.
    - - A change in the size or position of a figure or graph of the function is called a transformation.
    - - Rigid transformations
        
        Vertical Translations
        
        Appearance in Function
        
        Transformation of Graph
        
        Transformation of Point
        
        (𝒙,𝒚)→(𝒙,𝒚+𝒂)
        
        (𝒙,𝒚)→(𝒙,𝒚−𝒂)
        
        𝒂 𝑢𝑛𝑖𝑡𝑠 𝑢𝑝
        
        𝒂 𝑢𝑛𝑖𝑡𝑠 𝑑𝑜𝑤𝑛
        
        𝒇(𝒙)→𝒇(𝒙)+𝒂
        
        𝒇(𝒙)→𝒇(𝒙)−𝒂
        
        Horizontal Translations
        
        Appearance in Function
        
        Transformation of Graph
        
        Transformation of Point
        
        (𝒙,𝒚)→(𝒙+𝒃,𝒚)
        
        (𝒙,𝒚)→(𝒙−𝒃,𝒚)
        
        𝒃 𝑢𝑛𝑖𝑡𝑠 𝑟𝑖𝑔ℎ𝑡
        
        𝒃 𝑢𝑛𝑖𝑡𝑠 𝑙𝑒𝑓𝑡
        
        𝒇(𝒙)→𝒇(𝒙−𝒃)
        
        𝒇(𝒙)→𝒇(𝒙+𝒃)
        
        Reflections in x-axes
        
        Appearance in Function
        
        Transformation of Graph
        
        Transformation of Point
        
        (𝒙,𝒚)→(𝒙,−𝒚)
        
        It is a symetry
        
        𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑒𝑑 𝑖𝑛 𝑡ℎ𝑒 𝒙 𝒂𝒙𝒊𝒔
        
        𝒇(𝒙)→−𝒇(𝒙)
        
        Reflections in y-axes
        
        Appearance in Function
        
        Transformation of Graph
        
        Transformation of Point
        
        (𝒙,𝒚)→(−𝒙,𝒚)
        
        It is a symetry
        
        𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑒𝑑 𝑖𝑛 𝑡ℎ𝑒 𝒚 𝒂𝒙𝒊𝒔
        
        𝒇(𝒙)→𝒇(−𝒙)
      - Non rigid transformations
        
        Horizontal Dilations
        
        Appearance in Function
        
        Transformation of Graph
        
        Transformation of Point
        
        (𝒙,𝒚)→(𝒙/𝒅,𝒚)
        
        (𝒙,𝒚)→(𝒙/𝒅,𝒚)
        
        𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑒𝑑
        ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙𝑙𝑦
        
        𝑒𝑥𝑝𝑎𝑛𝑑𝑒𝑑
        ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙𝑙𝑦
        
        𝒇(𝒙)→𝒇(𝒅𝒙)
        𝒅>𝟏
        
        𝒇(𝒙)→𝒇(𝒅𝒙)
        𝟎<𝒅<𝟏
        
        Since x/d is the formula, if we put in the value of 2, we will narrow it down by dividing 2 by the value of x.
        
        Since x/d is the formula, we will expand the formula by multiplying x by 2 if we put in ½.
        
        Vertical Dilations
        
        Appearance in Function
        
        Transformation of Graph
        
        Transformation of Point
        
        (𝒙,𝒚)→(𝒄𝒙,𝒚)
        
        (𝒙,𝒚)→(𝒄𝒙,𝒚)
        
        𝑒𝑥𝑝𝑎𝑛𝑑𝑒𝑑
        𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙𝑙𝑦
        
        𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑒𝑑
        𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙𝑙𝑦
        
        𝒇(𝒙)→𝒄𝒇(𝒙)
        𝒄>𝟏
        
        𝒇(𝒙)→𝒄𝒇(𝒙)
        𝟎<𝒄<𝟏
        
        In the first, we increase y, so it expands.
        
        The reason why we multiply x with a coefficient is that we need to increase x since y will increase anyway so that the equality in the function does not break.
        
        We scale down proportionally
        
        Because we always divide x value because we multiply by values like ½. So the value is getting smaller.
      - Important note when reading the appearance in the Function
        
        When it's effecting the x-axis
        
        for example it looks like 𝒇(𝒙)→𝒇(𝒙−𝒃)
        
        it means that the added value will effect the x value directly. Do not confuse it with y. Which is in front of the equation.
        
        What do I mean?
        
        original version => f(x) = x + 4
        
        1 more item...
        
        the changed version=> f(x - 3)= x + 4
        
        1 more item...
        
        In other words the original x axis.
        
        the a value will change the x variable in the rule. When that happens the equality will change from the parent function.
        
        It means that "x" value will be changed.
        
        When it's effecting the y-axis
        
        for example it looks like f(x) = x +/ - a
        
        it means that the result will be equal to "y"
        
        What do I mean?
        
        y = x + or - a
        
        the a value will change the x variable. When that happens the equality will change from the parent function.
        
        It means that "y" value will be changed.
- - - - x (root 1) + x (root 2) = -b/a
      - x (root 1) * x (root 2) = c/a