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Determine whether the following functions are even or odd:

To determine whether the function \( f(x) = x \sqrt{x^2 + 5} \) is even, odd, or neither, we will use the following definitions:

1. **Even Function**: A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \).
2. **Odd Function**: A function \( f(x) \) is odd if \( f(-x) = -f(x) \) for all \( x \).

Let's test these properties.

### Step 1: Compute \( f(-x) \)
The given function is:
\[
f(x) = x \sqrt{x^2 + 5}
\]
Now, substitute \( -x \) into the function:
\[
f(-x) = (-x) \sqrt{(-x)^2 + 5} = (-x) \sqrt{x^2 + 5}
\]

### Step 2: Compare \( f(-x) \) with \( f(x) \)
We have:
\[
f(-x) = -x \sqrt{x^2 + 5}
\]
And:
\[
f(x) = x \sqrt{x^2 + 5}
\]

- \( f(-x) \neq f(x) \), so the function is **not even**.
- \( f(-x) = -f(x) \), which shows the function satisfies the condition for being **odd**.

### Conclusion:
The function \( f(x) = x \sqrt{x^2 + 5} \) is **odd**.

Would you like further clarification or details on this? Here are some related questions to explore this concept further:

1. What are some examples of even functions?
2. Can a function be neither even nor odd? If so, how can we identify it?
3. How would you check if a polynomial function is even or odd?
4. Can the square root function \( \sqrt{x} \) be even or odd?
5. What are the applications of even and odd functions in physics or engineering?

**Tip:** When determining whether a function is even or odd, always check both conditions \( f(-x) = f(x) \) and \( f(-x) = -f(x) \) systematically to avoid mistakes!

Explore whether the function f(x) = x sqrt(x^2 + 5) is even, odd, or neither with a detailed analysis. Learn the definitions of even and odd functions and how to apply them. Suitable for high school students and beyond.

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