A Function Invariant under a Group of Transformations
A Function Invariant under a Group of Transformations
The functions (x), , given by (x)=x, (x)=, (x)=1-x, (x)=, (x)=, and (x)= form a group with the composition of maps; for instance ((x))=(x).
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x
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1-x
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x-1
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x
x-1
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Moreover, if we take , we have , , that is, the function is invariant with respect to the group of transformations . The graph demonstrates this behaviour: choose a real number on the horizontal axis; the numbers (λ), are shown on the same axis and the corresponding values (red points) are all at the same height.
f(x)=(x-1)
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(-x+1)
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f((x))=f(x)
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f(x)
{(x),(x),…,(x)}
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λ
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f((λ))
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