Question

Find odd, even or neither symmatry using

odd = f (-x) = -f(x)
or
even = f(-x) = f(x)
a) f (x) = -2x^3 (3x-1)^2 (3x + 1)^2

Answer 1

The function -2x^3 (3x-1)^2 (3x + 1)^2 has odd symmetry.

Step-by-step explanation:

An even function is symmetric about the y-axis, which means that if you replace x with -x in the function and the function remains the same, then the function is even. An odd function is generated by reflecting the function about the y-axis and then about the x-axis, which means if you replace x with -x in the function and the function becomes the negative of the original, then the function is odd.

In the provided function f(x) = -2x^3 (3x-1)^2 (3x + 1)^2, we can check for even or odd symmetry. By replacing x with -x in the function, we get:
f(-x) = -2(-x)^3 (3(-x)-1)^2 (3(-x) + 1)^2
Simplifying this expression, we get:
f(-x) = -2x^3 (-3x-1)^2 (-3x + 1)^2

Now, we can compare f(-x) with -f(x):

-f(x) = -(-2x^3 (3x-1)^2 (3x + 1)^2) = 2x^3 (3x-1)^2 (3x + 1)^2

Since f(-x) = -2x^3 (-3x-1)^2 (-3x + 1)^2 is equal to -f(x) = 2x^3 (3x-1)^2 (3x + 1)^2, the function f(x) = -2x^3 (3x-1)^2 (3x + 1)^2 has odd symmetry.