Question

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).

Answer 1

B.

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).

Answer 2

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

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