Check Please?

Determine if the given function is even, odd or neither.

1. f(x) = 3x^4 - 2x^2

=Even.

2. f(x) = x^3 + x

= Odd.

correct

To determine if a given function is even, odd, or neither, we need to analyze its symmetry with respect to the y-axis or origin.

For a function to be even, it must satisfy the property f(x) = f(-x) for all values of x. This means that if we substitute -x for x in the function, we should get an identical expression. When a function satisfies this condition, its graph is symmetric with respect to the y-axis.

On the other hand, for a function to be odd, it must satisfy the property f(x) = -f(-x) for all values of x. This means that if we substitute -x for x in the function, we should get the same expression with its sign reversed. When a function satisfies this condition, its graph is symmetric with respect to the origin.

Let's apply these properties to the given functions:

1. f(x) = 3x^4 - 2x^2

To check if this function is even, we substitute -x for x and see if the expression remains the same:
f(-x) = 3(-x)^4 - 2(-x)^2
= 3x^4 - 2x^2

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

2. f(x) = x^3 + x

To check if this function is odd, we substitute -x for x and see if the expression changes sign:
-f(-x) = -(x^3 + x)
= -x^3 - x

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

Therefore, the given functions are:

1. f(x) = 3x^4 - 2x^2 -> Even.
2. f(x) = x^3 + x -> Odd.