Question

$f(x) = x^6 + x^3 + 4$
$f(x) = 3x^4 - 2x^2 + 1$
$f(x) = 2x^5 + x^4 - 3x$
$f(x) = x^2 + 6x + 3$
$f(x) = 5x^3 - x$
$f(x) = 3x^7 + x^3 - 2x$
$f(x) = 4x^4 - x^2 + 7$
$f(x) = x^6 - 3x^3 + 2x$
$f(x) = 2x^8 + x^4$
$f(x) = 5x^5 + 3x^2 + 1$
$f(x) = 3x^3 - 2x^2 + 4$

Explanation:

Since the problem is not clearly stated (e.g., what to do with these functions: classify as even/odd, find degree, etc.), we need more details. But assuming a common task like identifying even functions (symmetric about y - axis, \(f(-x)=f(x)\)):

Step1: Recall even function definition

A function \(y = f(x)\) is even if \(f(-x)=f(x)\) for all \(x\) in the domain. This means that the function has only even powers of \(x\) (or constant term, which is \(x^0\)).

Step2: Check each function

• \(f(x)=x^6 + x^3+4\): Has \(x^3\) (odd power), not even.
• \(f(x)=3x^4 - 2x^2 + 1\): All powers (\(x^4,x^2,x^0\)) are even. Let's verify: \(f(-x)=3(-x)^4-2(-x)^2 + 1=3x^4-2x^2 + 1=f(x)\). So it's even.
• \(f(x)=2x^5+x^4 - 3x\): Has \(x^5\) and \(x\) (odd powers), not even.
• \(f(x)=x^2 + 6x + 3\): Has \(x\) (odd power), not even.
• \(f(x)=5x^3 - x\): Odd powers, not even.
• \(f(x)=3x^7+x^3 - 2x\): Odd powers, not even.
• \(f(x)=4x^4 - x^2 + 7\): All powers (\(x^4,x^2,x^0\)) are even. \(f(-x)=4(-x)^4-(-x)^2 + 7=4x^4 - x^2 + 7=f(x)\). Even.
• \(f(x)=x^6-3x^3 + 2x\): Has \(x^3\) and \(x\) (odd powers), not even.
• \(f(x)=2x^8+x^4\): All powers (\(x^8,x^4\)) are even. \(f(-x)=2(-x)^8+(-x)^4=2x^8 + x^4=f(x)\). Even.
• \(f(x)=5x^5+3x^2 + 1\): Has \(x^5\) (odd power), not even.
• \(f(x)=3x^3-2x^2 + 4\): Has \(x^3\) (odd power), not even.

Answer:

If the task is to find even functions, the answers are \(f(x)=3x^4 - 2x^2 + 1\), \(f(x)=4x^4 - x^2 + 7\), \(f(x)=2x^8+x^4\) (depending on the actual question, since the original problem's task is not specified, this is a common analysis for such a list of functions).