To determine whether a function is even, odd, or neither, we use the following tests:

• Even Function: $f(x)$ is even if $f(-x) = f(x)$ for all $x$.
• Odd Function: $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$.
• Neither: If neither of the above conditions hold, the function is neither even nor odd.

Now, let's analyze each function:

1. $f(x) = x^2 + 4$

• To check if it's even: $f(-x) = (-x)^2 + 4 = x^2 + 4$ Since $f(-x) = f(x)$, the function is even.

2. $g(x) = x^3 + 4$

• To check if it's odd: $g(-x) = (-x)^3 + 4 = -x^3 + 4$ $g(-x) \neq -g(x)$ and $g(-x) \neq g(x)$, so the function is neither.

3. $h(x) = 4x^3 + x$

• To check if it's odd: $h(-x) = 4(-x)^3 + (-x) = -4x^3 - x = -(4x^3 + x)$ Since $h(-x) = -h(x)$, the function is odd.

Final answers:

• $f(x) = x^2 + 4$ is even.
• $g(x) = x^3 + 4$ is neither.
• $h(x) = 4x^3 + x$ is odd.

Would you like any further details or explanations?

Relative Questions:

1. What is the geometric interpretation of an even function?
2. Can a function be both even and odd? Why or why not?
3. How do you determine if a function is neither even nor odd?
4. What kind of symmetry does an odd function exhibit on a graph?
5. How do polynomials' degrees affect whether they are even, odd, or neither?

Tip: If a function contains only even powers of $x$, it tends to be even, and if it contains only odd powers, it tends to be odd.