The image contains four questions asking whether each given function is odd, even, or neither.

Let's analyze each function one by one:

1. Function 1: $f(x) = \frac{x^3 - 1}{x}$

• Odd Function: A function $f(x)$ is odd if $f(-x) = -f(x)$.
• Even Function: A function $f(x)$ is even if $f(-x) = f(x)$.

Calculating $f(-x)$:

$f(-x) = \frac{(-x)^3 - 1}{-x} = \frac{-x^3 - 1}{-x} = \frac{-x^3 - 1}{-x} = \frac{x^3 + 1}{x}$

This is not equal to $f(x)$ or $-f(x)$. Therefore, the function is neither even nor odd.

2. Function 2: $g(x) = x(x + 1)$

Calculating $g(-x)$:

$g(-x) = (-x)((-x) + 1) = (-x)(-x + 1) = x(x - 1)$

Since $g(-x) \neq g(x)$ and $g(-x) \neq -g(x)$, the function is neither even nor odd.

3. Function 3: $h(x) = \sqrt{1 - x^2}$

Calculating $h(-x)$:

$h(-x) = \sqrt{1 - (-x)^2} = \sqrt{1 - x^2}$

Since $h(-x) = h(x)$, the function is even.

4. Function 4: $k(x) = 4x^3$

Calculating $k(-x)$:

$k(-x) = 4(-x)^3 = 4(-x^3) = -4x^3$

Since $k(-x) = -k(x)$, the function is odd.

Conclusion:

1. Function 1 is neither.
2. Function 2 is neither.
3. Function 3 is even.
4. Function 4 is odd.

Would you like further details on any of these explanations, or do you have any questions?

Here are some related questions to deepen your understanding:

1. What are the conditions for a function to be even or odd?
2. How can we graphically determine if a function is even or odd?
3. What are some examples of functions that are both even and odd?
4. What happens to the symmetry of a function if we add or multiply by another function?
5. Can a function that is neither even nor odd be split into a sum of an even and an odd function?

Tip: Checking the symmetry of a function is often easiest by substituting $x$ with $-x$ and comparing the result to the original function.