is X cubed symmetric about the x-axis, Y axis, the origin, or none?
To determine if the function f(x) = x^3 is symmetric about the x-axis, y-axis, or the origin, we need to consider how the function behaves when certain changes are made to the input variable x.
1. Symmetry about the x-axis:
To test if the function is symmetric about the x-axis, we need to see if replacing x with -x yields the same function value. Let's substitute -x into the function:
f(-x) = (-x)^3 = -x^3
Since f(-x) = -x^3, which is not equal to f(x) = x^3, we can conclude that the function is not symmetric about the x-axis.
2. Symmetry about the y-axis:
To test if the function is symmetric about the y-axis, we need to see if replacing x with -x yields the same function value as when x is used. Let's substitute -x into the function:
f(-x) = (-x)^3 = -x^3
Since f(-x) = -x^3, which is equal to f(x) = x^3, we can conclude that the function is symmetric about the y-axis.
3. Symmetry about the origin (0,0):
To test if the function is symmetric about the origin, we need to see if replacing x with -x yields the negative of the function value. Let's substitute -x into the function:
f(-x) = (-x)^3 = -x^3
We can see that f(-x) = -x^3, which is the negative of f(x) = x^3.
Therefore, we can conclude that the function f(x) = x^3 is symmetric about the y-axis and the origin, but not symmetric about the x-axis.