A student says that the function f (x) = 3x^4+5x^2+1 is an even function.

Is the student's statement true or not true, and why?

The student's claim is true, because for any input of x, f (x) = -f (x).

The student's claim is true, because for any input of x, f (x) = f (-x).

The student's claim is not true, because for any input of x, f (x) = f (-x).

The student's claim is not true, because for any input of x, f (x) = -f (x).

The student's claim is true, because for any input of x, f (x) = f (-x).

Step-by-step explanation:

If a student says that the function f (x) = 3x^4+5x^2+1 is an even function, the student's statement true because for any input of x, f (x) = f (-x).

f (-x) = f (x) is even.

f (-x) = -f (x) is odd

B.

Step-by-step explanation:

If f (-x) = f (x), then f is even.

If f (-x) = -f (x), then f is odd.

To determine if f (x) = 3x^4+5x^2+1 is even or odd plug in - x like so:

f (x) = 3x^4+5x^2+1

f (-x) = 3 (-x) ^4+5 (-x) ^2+1

f (-x) = 3x^4+5x^2+1

f (-x) = f (x)

So f is even.

You should keep in mind the following:

(-x) ^odd = - (x^odd)

(-x) ^even=x^even

Examples:

(-x) ^81 = - (x^81) since 81 is odd

(-x) ^10=x^10 since 10 is even

Anyways, the student is right and f (-x) = f (x).