https://file.aiquickdraw.com/mathbot/file/17265376946098etc4frv.jpg

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.

In Exercises 47-54, state whether the function is odd, even, or neither for the following functions: f(x) = 2x^4, g(x) = x^3, f(x) = sqrt(x^2 + 2), g(x) = 3 / (1 + x^2), f(x) = -x^2 + 0.03x + 5, f(x) = x^3 + 0.04x^2 + 3, g(x) = 2x^3 - 3x, h(x) = 1/x.

Learn how to determine if a function is odd, even, or neither with a detailed analysis of functions from Exercises 47-54. This solution covers key concepts in algebra and symmetry of functions, using formulas for even and odd functions. Suitable for high school students in Grades 10-12.

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