Question

for each function find f(-x) and -f(x) and then determine whether it is even odd or neither f(x)=2x^3+1/x

Answer 1

Function is odd.

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

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

Explanation:

f(-x) -> f(x) = 2(-x)^3+ (1/-x) which equals -2x^3 - 1/x.

-f(x) = -2x^3- 1/x.

Since f(x) doesn't equal f(-x), the function isn't even.

Since f(-x)=-f(x), the function is odd.

Hope this helps have a great day!

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Answer 2

To find f(-x), we substitute -x for x in the given function:

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

Simplifying,

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

To find -f(x), we negate the entire function:

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

= -2x^3 - 1/x

Now let's determine whether the function is even, odd, or neither.

A function is even if f(x) = f(-x) for all values of x. In this case, we can see that f(-x) = -2x^3 - 1/x, which is not equal to f(x) = 2x^3 + 1/x. Therefore, the function is not even.

A function is odd if -f(x) = f(-x) for all values of x. In this case, we can see that -f(x) = -(-2x^3 - 1/x) = 2x^3 + 1/x. Similarly, f(-x) = -2x^3 - 1/x. We can observe that -f(x) = f(-x), so the function is odd.

Therefore, the given function f(x) = 2x^3 + 1/x is odd.