To determine whether each function given in the image is even, odd, or neither, we'll analyze each function individually. A function $f(x)$ is:

• Even if $f(-x) = f(x)$ for all $x$ in the domain.
• Odd if $f(-x) = -f(x)$ for all $x$ in the domain.
• Neither if it does not satisfy either of these conditions.

Let's analyze each function from the image:

1. $f(x) = 2x + 1$

   • $f(-x) = 2(-x) + 1 = -2x + 1$
   • $f(-x) \neq f(x)$ and $f(-x) \neq -f(x)$, so the function is neither.

2. $f(x) = 3x^2 + 2x - 1$

   • $f(-x) = 3(-x)^2 + 2(-x) - 1 = 3x^2 - 2x - 1$
   • $f(-x) \neq f(x)$ and $f(-x) \neq -f(x)$, so the function is neither.

3. $f(x) = \sqrt{x - 1}$

   • $f(-x) = \sqrt{-x - 1}$
   • The domain of $f(x) = \sqrt{x - 1}$ is $x \geq 1$, so $-x - 1$ would not be valid unless $x \leq -1$, leading to different domains. Thus, the function is neither.

4. $f(x) = \sqrt{x^2 + 1}$

   • $f(-x) = \sqrt{(-x)^2 + 1} = \sqrt{x^2 + 1} = f(x)$
   • Since $f(-x) = f(x)$, the function is even.

5. $f(x) = \sqrt{x} + 1$

   • $f(-x)$ is not defined for real numbers since the square root of a negative number is not real. Thus, the function is neither.

6. $f(x) = \frac{1}{\sqrt{1 - x^2}}$

   • $f(-x) = \frac{1}{\sqrt{1 - (-x)^2}} = \frac{1}{\sqrt{1 - x^2}} = f(x)$
   • Since $f(-x) = f(x)$, the function is even.

7. $f(x) = -\frac{x}{\sqrt{2x - 3}}$

   • $f(-x) = -\frac{-x}{\sqrt{2(-x) - 3}} = \frac{x}{\sqrt{-2x - 3}}$
   • $f(-x) \neq f(x)$ and $f(-x) \neq -f(x)$, so the function is neither.

8. $f(x) = \sqrt{2x}$

   • $f(-x) = \sqrt{2(-x)} = \sqrt{-2x}$
   • $f(-x) \neq f(x)$ and $f(-x) \neq -f(x)$, so the function is neither.

9. $f(x) = 3x - \sqrt{2}$

   • $f(-x) = 3(-x) - \sqrt{2} = -3x - \sqrt{2}$
   • $f(-x) \neq f(x)$, but $f(-x) = -f(x)$, so the function is odd.

10. $f(x) = \frac{3x - 1}{x + 2}$

    • $f(-x) = \frac{3(-x) - 1}{-x + 2} = \frac{-3x - 1}{-x + 2}$
    • $f(-x) \neq f(x)$ and $f(-x) \neq -f(x)$, so the function is neither.

Summary of Functions:

1. Neither
2. Neither
3. Neither
4. Even
5. Neither
6. Even
7. Neither
8. Neither
9. Odd
10. Neither

Would you like further explanations on any of these functions or details on how the even/odd properties work?

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Here are 5 follow-up questions to expand your understanding:

1. What are the implications of a function being even or odd in real-world applications?
2. Can a function be both even and odd? If yes, what would that function look like?
3. How does the domain of a function affect whether it can be classified as even or odd?
4. What are some common examples of even and odd functions in physics or engineering?
5. How would you graphically verify if a function is even or odd?

Tip: A good way to check if a function is even or odd is to substitute $x$ with $-x$ and compare the resulting expression to the original function.