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IGCSE Additional Mathematics - Coggle Diagram

- - - - Range of Inputs
      - x > 0
    - - Range of Possible Outputs :
      - f(x) > -1
  - - - Inverse Functions
        
        f(x) = 2x - 1 | x = 2y - 1 | 2y = x + 1 | y = ( x + 1 )/2 | f'(x) = ( x + 1 )/2
      - Modulus Function
        
        Aka Absolute Value or Magnitude of the Value
        
        Denoted by ||
        
        |-21| = 21
      - Composite Functions
        
        f(x)=2x , g(x)=3x | fg(x)= 2(3x) = 6x
    - - Many-to-One Functions
        
        Quadratic Functions
        
        f(x) = x^2 - 1
        
        Minimum/Maximum Points
        
        X intercepts at ( 0 , 2 ) , ( 0 , 3 ) | Minimum/Maximum point's y=2.5 as ( 3 + 2 )/2 = 2.5
        
        Minimum if x^2 is positive | Maximum if x^2 is negative
        
        Methods to Solve Quadratic Functions
        
        Factorisation
        
        x^2 -2x + 1 = 0 | ( x - 1 )^2 = 0 | x = 1
        
        Completing the Square
        
        x^2 + 7x -17 = 0 | ( x^2 + 7x + 49/4 ) - 17 = 49/4 | ( x + 7/2 )^2 = 117/4 | x = +- √(117/4) - 7/2 | x = 1.91 , -8.91
        
        Quadratic Equation
        
        (-b +- √(-b^2 - 4ac))/2a
        
        ?
        
        x^4 - x^2 +1/2 = 0 | u=x^2 and u^2 - u + 1/2 = 0
        
        Quadratic Inequalities
        
        x^2 - x + 1/4 > 0 | ( x - 1/2 )^2 > 0 | x = 1/2
        
        Finding the number of roots in a quadratic function
        
        Discriminant
        
        Uses
        
        Finding no of roots
        
        3 more items...
        
        Finding unknown value within quadratic functions
        
        Types of Quadratic Functions
        
        Complete
        
        f(x) = x^2 - x + 1
        
        Missing constant?
        
        2x^2 + 2x = 0 | 2x ( x + 1 ) = 0 | x = 0 , -1
        
        Missing middle term
        
        x^2 - 1 = 0 | x^2 = 1 | x = +-1
        
        x^2 + 1 = 0 | x^2 = -1 | x = n/a
        
        Modulus Functions
        
        Modulus is denoted by ||, aka Absolute Value or Magnitude of a Value
        
        | x + 1 | = 2 // x + 1 = 2 and x + 1 = -2
        
        Modulus Inequalities
        
        | x + 1 | > 2 // x + 1 > 2 and x + 1 < -2
        
        If both sides are constrained by a modulus, the only the answer when bot sides are positive, and one side is negative should be considered
        
        Cubic Functions ( That have 2 / 3 unique roots )
      - One-to-One Functions
        
        f(x) = cos x where 0 <= x <= 180
        
        f(x) = n^x where n>=0
        
        f(x) = n^(2x-1) where x is an integer
        
        Linear Functions
        
        f(x) = 2x - 1