https://file.aiquickdraw.com/mathbot/file/172556766692364pwfd1m.jpg

It appears you've uploaded an image with a mathematical problem involving determining whether certain rational functions are **even**, **odd**, or **neither**. I will transcribe and solve the functions for you.

The problem asks for the classification of functions based on their symmetry:
- **Even**: A function is even if \( f(-x) = f(x) \).
- **Odd**: A function is odd if \( f(-x) = -f(x) \).
- **Neither**: If neither condition is satisfied, the function is neither even nor odd.

Let me analyze the functions from the image and determine their nature.

### List of functions:
1. \( f(x) = \frac{2x^4 - 5}{2x^2 - 4x^2} \)
2. \( f(x) = \frac{x^2}{7x^2 + 1} \)
3. \( f(x) = \frac{4}{x^3 - 7} \)
4. \( f(x) = \frac{x^2 - 1}{5x} \)
5. \( f(x) = \frac{2x^3 + 1}{x} \)
6. \( f(x) = \frac{x^2 - 7}{5} \)
7. \( f(x) = \frac{2x^2 + 7}{x^2 - 1} \)

Let me work through them to classify each as even, odd, or neither.Here are the classifications of the given functions based on their symmetry:

1. \( f(x) = \frac{2x^4 - 5}{2x^2 - 4x^2} \): **Even**
2. \( f(x) = \frac{x^2}{7x^2 + 1} \): **Even**
3. \( f(x) = \frac{4}{x^3 - 7} \): **Neither**
4. \( f(x) = \frac{x^2 - 1}{5x} \): **Odd**
5. \( f(x) = \frac{2x^3 + 1}{x} \): **Neither**
6. \( f(x) = \frac{x^2 - 7}{5} \): **Even**
7. \( f(x) = \frac{2x^2 + 7}{x^2 - 1} \): **Even**

Would you like any details on how these were determined, or do you have any specific questions?

Here are some related questions to deepen understanding:
1. What is the graphical significance of even and odd functions?
2. How can we derive if a polynomial function is even or odd by just inspecting its degree?
3. Can a function be both even and odd? If so, which functions fall into this category?
4. How does symmetry affect the integration of even and odd functions over symmetric intervals?
5. How do transformations such as shifting and scaling affect the even or odd nature of a function?

**Tip**: Even functions have symmetry about the y-axis, while odd functions have rotational symmetry about the origin.

Determine whether each of the following rational functions is even, odd, or neither.

Determine whether rational functions are even, odd, or neither using symmetry properties. Learn the conditions for even and odd functions with worked examples for Grades 10-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation