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The problem in your image asks you to fill out the table of values for three functions and determine whether each function is even, odd, or neither. Here's a breakdown of the functions provided:

1. \( f(x) = -x^6 + 3x^2 \)
2. \( f(x) = \sin(x) + 3 \)
3. \( f(x) = \tan(x) \)

To determine if a function is **even**, **odd**, or **neither**, we need to follow these rules:
- A function is **even** if \( f(-x) = f(x) \) for all \(x\).
- A function is **odd** if \( f(-x) = -f(x) \) for all \(x\).
- If neither condition holds, then the function is neither.

### Let's start by analyzing the functions:

#### 1. \( f(x) = -x^6 + 3x^2 \)
This is a polynomial function, and we can determine its symmetry by testing \( f(-x) \):
- \( f(-x) = -(-x)^6 + 3(-x)^2 = -x^6 + 3x^2 \)
Since \( f(-x) = f(x) \), the function is **even**.

#### 2. \( f(x) = \sin(x) + 3 \)
For trigonometric functions, recall:
- \( \sin(-x) = -\sin(x) \)
So,
- \( f(-x) = \sin(-x) + 3 = -\sin(x) + 3 \)
Since \( f(-x) \neq f(x) \) and \( f(-x) \neq -f(x) \), this function is **neither**.

#### 3. \( f(x) = \tan(x) \)
For the tangent function, recall:
- \( \tan(-x) = -\tan(x) \)
Thus,
- \( f(-x) = \tan(-x) = -\tan(x) \)
Since \( f(-x) = -f(x) \), the function is **odd**.

### Now you can complete the table

Fill out the table of values for three functions: f(x) = -x^6 + 3x^2, f(x) = sin(x) + 3, and f(x) = tan(x), and determine if each function is even, odd, or neither.

Understand how to classify polynomial and trigonometric functions as even, odd, or neither through step-by-step analysis of f(x) = -x^6 + 3x^2, f(x) = sin(x) + 3, and f(x) = tan(x). Ideal for high school students, Grades 10-12.

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