testing symmetrytest the symmetry f(x) = x^5 + x^3 + x

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justaguide | College Teacher | (Level 2) Distinguished Educator

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Symmetrical functions are those that are odd. If the given function is odd, it is symmetrical.

f(x) = x^5 + x^3 + x

f(-x) = (-x)^5 +(-x)^3 +(-x)

=> -x^5 - x^3 - x

=> -( x^5 + x^3 + x)

=> -f(x)

This is the definition of an odd function, f(-x) = -f(x). Therefore it is symmetrical.

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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To test the symmetry of the function, we'll have to look at f(−x):

f(-x) = (-x)^5 + (-x)^3 + (-x)

A negative number raised to an odd power, yields a negative result.

f(-x) = -x^5  - x^3 - x

We'll factorize by -1:

f(-x) = -(x^5 + x^3 + x)

f(-x) = -f(x)

Since  f(−x) = −f(x), the function is symmetrical with respect to the origin.

A function that is symmetrical with respect to the origin is an odd function.

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