What are some examples of a symmetric function?

Answer 1

#f(x, y) = x^2 + xy + y^2#
#g(x, y, z) = xy + yz + zx + 1/(x^2+y^2+z^2)#

A symmetric function is a function in several variable which remains unchanged for any permutation of the variables.

For example, if #f(x, y) = x^2 + xy + y^2#, then #f(y, x) = f(x, y)# for all #x# and #y#.
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Answer 2

Some examples of symmetric functions include:

  1. Even functions: Functions where ( f(x) = f(-x) ) for all ( x ) in the domain. Examples include ( f(x) = x^2 ) and ( f(x) = \cos(x) ).

  2. Odd functions: Functions where ( f(x) = -f(-x) ) for all ( x ) in the domain. Examples include ( f(x) = x^3 ) and ( f(x) = \sin(x) ).

  3. Functions symmetric about the y-axis: Functions where ( f(x) = f(-x) ) for all ( x ) in the domain. Examples include ( f(x) = |x| ) and ( f(x) = x^4 - 2x^2 + 1 ).

These functions exhibit various types of symmetry that can be visualized graphically or algebraically.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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