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Year 12 Maths Specialist Topics 3.2.1 - 3.2.5: Functions - Coggle Diagram

- - - - A function's inverse reverses the process that the function did.
        eg if we double something then the inverse is to halve it.
        
        Inverse can be found
        
        algebraically
        
        Replace f(x) with y
        
        Swap x and y
        
        Rearrange to make y the subject
        
        Replace y with f^-1(x)
        
        to check/prove
        
        Inverse Function Property (IFP):
        fof-1 :(x)=x
        (function of inverse function of x equals x, AND inverse function of function of x equals x)
        
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        https://www.youtube.com/watch?v=TN4ybFiuV3k&ab_channel=TheOrganicChemistryTutor
        
        graphically
        
        The inverse of a function is its mirror about the line y=x (the identity function)
        
        numerically
        
        Any cartesian point (x,y) becomes (y,x) eg (2,1) --> (1,2), (45,5)-->(5,45)
    - - these are called
        
        Composite functions
        
        written as
        
        f(g(x)) or g(f(x))
        
        "Is the range of the inner function a subset of the domain of the outer function?"
        
        YES
        
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        NO
        
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        may be
        
        undefined
        
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        defined
        
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        Work from the inside out - solve internal function before external
        
        EXAMPLE #1
        
        https://www.youtube.com/watch?v=ZFPkQkURSxk&ab_channel=TheOrganicChemistryTutor
    - - Input= Domain
        
        examples
        
        all real numbers
        
        all positive numbers
        
        all numbers in a given range
      - Output = Range
        
        examples
        
        all real numbers
        
        all positive numbers
        
        all numbers in a given range
  - - - example of a function
        
        Types of function
        
        one-to one
        vs
        many to one
        functions
        
        one to one
        
        tan function
        
        hyperbola
        
        explanation
        
        only one value of x gives any value of y
        
        examples
        
        linear equation
        
        many to one
        
        examples
        
        cosine function
        
        parabola
        
        explanation
        
        more than one value of x gives the same value of y
        
        sine function
        
        can be determined
        
        algebraically
        
        https://www.youtube.com/watch?v=HaxH1OmmIq0&ab_channel=SkanCityAcademy
        
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        For any f(x), let f(a)=f(b). If a=b the function is one-to-one. If a=/=b, it is not a one-to-one function.
        
        graphically
        
        Horizontal line test
        
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        numerically
        
        eg
        
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        defined
        vs
        undefined
        functions
        
        https://www.youtube.com/watch?v=sS4PWFvgPiE&ab_channel=TravisCollier
        
        EXAMPLE #4
      - example: not a function
        
        https://byjus.com/question-answer/what-is-a-function-rule/
        
        https://www.geeksforgeeks.org/types-of-functions/
      - Resource - geogebra tutorial:
        https://www.geogebra.org/m/ussndp3t