Is the function #f(x) = x^3# symmetric with respect to the y-axis?

Answer 1

No, it has rotational symmetry of order #2# about the origin.

An even function is a function satisfying: #f(-x) = f(x)" "# for all #x# in the domain of #f(x)color(white)(0/0)#
An odd function is a function satisfying: #f(-x) = -f(x)" "# for all #x# in the domain of #f(x)color(white)(0/0)#
Even functions are symmetric with respect to the #y#-axis.
Odd functions have rotational symmetry of order #2# about the origin.

Given:

#f(x) = x^3#
Note that for any value of #x#:
#f(-x) = (-x)^3 = (-1)^3 x^3 = -x^3 = -f(x)#
So #f(x) = x^3# is an odd function.
It is not symmetric with respect to the #y#-axis, but it has rotational symmetry of order #2# about the origin.

graph{x^3 [-5, 5, -10, 10]}

In fact any polynomial consisting of only terms of odd degree will be an odd function and any polynomial consisting of only terms of even degree will be an even function.

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Answer 2

No, the function (f(x) = x^3) is not symmetric with respect to the y-axis.

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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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