To determine whether a function is odd, even, or neither, we follow these steps:

1. Even Function: A function $f(x)$ is even if: $f(-x) = f(x)$
2. Odd Function: A function $f(x)$ is odd if: $f(-x) = -f(x)$
3. If neither condition holds, the function is neither even nor odd.

Let's go through each function from the image:

47. $f(x) = 2x^4$

• $f(-x) = 2(-x)^4 = 2x^4$
• Since $f(-x) = f(x)$, the function is even.

48. $g(x) = x^3$

• $g(-x) = (-x)^3 = -x^3$
• Since $g(-x) = -g(x)$, the function is odd.

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

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

50. $g(x) = \frac{3}{1 + x^2}$

• $g(-x) = \frac{3}{1 + (-x)^2} = \frac{3}{1 + x^2}$
• Since $g(-x) = g(x)$, the function is even.

51. $f(x) = -x^2 + 0.03x + 5$

• $f(-x) = -(-x)^2 + 0.03(-x) + 5 = -x^2 - 0.03x + 5$
• Since $f(-x) \neq f(x)$ and $f(-x) \neq -f(x)$, the function is neither even nor odd.

52. $f(x) = x^3 + 0.04x^2 + 3$

• $f(-x) = (-x)^3 + 0.04(-x)^2 + 3 = -x^3 + 0.04x^2 + 3$
• Since $f(-x) \neq f(x)$ and $f(-x) \neq -f(x)$, the function is neither even nor odd.

53. $g(x) = 2x^3 - 3x$

• $g(-x) = 2(-x)^3 - 3(-x) = -2x^3 + 3x = -(2x^3 - 3x)$
• Since $g(-x) = -g(x)$, the function is odd.

54. $h(x) = \frac{1}{x}$

• $h(-x) = \frac{1}{-x} = -\frac{1}{x}$
• Since $h(-x) = -h(x)$, the function is odd.

Summary:

• Even: 47, 49, 50
• Odd: 48, 53, 54
• Neither: 51, 52

Would you like more details or have any further questions?

Here are some related questions to consider:

1. What are the properties of even functions in terms of their graphs?
2. How do odd functions behave with respect to the origin?
3. What are some real-life applications of even and odd functions?
4. Can a function be both even and odd? Why or why not?
5. How would adding a constant to an even or odd function change its classification?

Tip: Remember that even functions are symmetric about the y-axis, while odd functions are symmetric about the origin.