Let's analyze each function $f(x)$, $g(x)$, and $h(x)$ to determine whether they are odd, even, or neither.

Steps to classify functions:

1. Even functions: A function is even if $f(-x) = f(x)$ for all $x$. Graphically, the function is symmetric about the y-axis.
2. Odd functions: A function is odd if $f(-x) = -f(x)$ for all $x$. Graphically, the function is symmetric about the origin.
3. Neither: If a function is neither even nor odd, it doesn't satisfy either condition.

1. $f(x) = 3x^3 + x$

• Find $f(-x)$: $f(-x) = 3(-x)^3 + (-x) = 3(-x^3) - x = -3x^3 - x$
• Check if it's odd or even:
  • $f(-x) = -(3x^3 + x) = -f(x)$, so $f(x)$ is odd.

2. $g(x) = \sqrt{x}$

• Find $g(-x)$: $g(-x) = \sqrt{-x}$ The square root of a negative number is undefined in the set of real numbers. So $g(x)$ doesn't satisfy either condition for being even or odd.
  • Therefore, $g(x)$ is neither.

3. $h(x) = 3 - x^2$

• Find $h(-x)$: $h(-x) = 3 - (-x)^2 = 3 - x^2$
  • Since $h(-x) = h(x)$, the function is even.

Summary:

• $f(x) = 3x^3 + x$ is odd.
• $g(x) = \sqrt{x}$ is neither.
• $h(x) = 3 - x^2$ is even.

Would you like a more detailed breakdown of any function, or do you have any questions?

Related Questions:

1. How can you graphically identify if a function is odd or even?
2. Why does the function $g(x) = \sqrt{x}$ not fit into either category?
3. Can a function be both even and odd? Why or why not?
4. How does the degree of a polynomial influence whether it's odd or even?
5. What happens if you try to apply these properties to piecewise functions?

Tip:

If you are ever unsure whether a function is odd, even, or neither, always test $f(-x)$ and compare it to both $f(x)$ and $-f(x)$.