To determine whether each function is odd, even, or neither, we use the following criteria:

• A function $f(x)$ is even if $f(-x) = f(x)$ for all $x$.
• A function $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$.
• Otherwise, the function is neither.

Solutions:

(a) $f(x) = x + x^2$

• Compute $f(-x) = -x + (-x)^2 = -x + x^2$.
• Since $f(-x) \neq f(x)$ and $f(-x) \neq -f(x)$, this function is neither.

(b) $f(x) = x + x^3$

• Compute $f(-x) = -x + (-x)^3 = -x - x^3 = -(x + x^3)$.
• Since $f(-x) = -f(x)$, this function is odd.

(c) $f(x) = e^{2x}$

• Compute $f(-x) = e^{2(-x)} = e^{-2x}$.
• Since $f(-x) \neq f(x)$ and $f(-x) \neq -f(x)$, this function is neither.

(d) $f(x) = e^{x^2}$

• Compute $f(-x) = e^{(-x)^2} = e^{x^2}$.
• Since $f(-x) = f(x)$, this function is even.

(e) $f(x) = x \sin x$

• Compute $f(-x) = (-x) \sin(-x) = -x (-\sin x) = -x \sin x = -f(x)$.
• Since $f(-x) = -f(x)$, this function is odd.

(f) $f(x) = x \cos x$

• Compute $f(-x) = (-x) \cos(-x) = -x \cos x$.
• Since $f(-x) \neq f(x)$ and $f(-x) \neq -f(x)$, this function is neither.

(g) $f(x) = |x|$

• Compute $f(-x) = |-x| = |x|$.
• Since $f(-x) = f(x)$, this function is even.

Summary:

• (a) Neither
• (b) Odd
• (c) Neither
• (d) Even
• (e) Odd
• (f) Neither
• (g) Even

Let me know if you'd like further explanations!

1. Why is symmetry important in determining whether a function is odd or even?
2. How do exponential functions relate to even or odd symmetry?
3. Can trigonometric functions combine to form odd or even functions?
4. What role does absolute value play in function symmetry?
5. How can graphing help in determining the nature of a function?

Tip: When unsure about a function’s symmetry, always test using $f(-x)$ and simplify