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