https://file.aiquickdraw.com/mathbot/file/1725504331926ok4uaejg.jpg

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.

Determine whether the following functions are even, odd, or neither:
1. f(x) = (x^3 - 1)/x
2. g(x) = x(x + 1)
3. h(x) = sqrt(1 - x^2)
4. k(x) = 4x^3

Explore how to determine if functions are even, odd, or neither using symmetry. Analyze functions like (x^3 - 1)/x, x(x + 1), sqrt(1 - x^2), and 4x^3 with step-by-step solutions. Suitable for students in Grades 9-12.

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