Determine whether the function f, of, x, equals, minus, 1, plus, 6, x, to the power 6 , minus, 9, x, squared*f*(*x*)=−1+6*x*6−9*x*2 is even, odd or neither.

To determine whether the function \( f(x) = -1 + 6x^6 - 9x^2 \) is even, odd, or neither, we use the same criteria as before:

1. **Even Function**: \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \).
2. **Odd Function**: \( f(x) \) is odd if \( f(-x) = -f(x) \) for all \( x \).
3. **Neither**: If neither condition is satisfied, the function is neither even nor odd.

### Step 1: Compute \( f(-x) \)

Given:
\[
f(x) = -1 + 6x^6 - 9x^2
\]

Compute \( f(-x) \):
\[
f(-x) = -1 + 6(-x)^6 - 9(-x)^2
\]

Since:
\[
(-x)^6 = x^6 \quad \text{and} \quad (-x)^2 = x^2,
\]

we have:
\[
f(-x) = -1 + 6x^6 - 9x^2
\]

### Step 2: Compare \( f(-x) \) with \( f(x) \)

Now we see that:
\[
f(-x) = -1 + 6x^6 - 9x^2 = f(x)
\]

### Conclusion

Since \( f(-x) = f(x) \), the function \( f(x) = -1 + 6x^6 - 9x^2 \) is an **even function**.

Would you like more details or have any questions? Here are some related questions:

1. What is the behavior of polynomials with only even powers of \(x\)?
2. Can a function be both even and odd? If not, why?
3. How does a constant term affect the evenness or oddness of a function?
4. What are some examples of odd polynomial functions?
5. How do transformations (like shifts or reflections) affect the evenness or oddness of a function?

**Tip:** For any polynomial, if all terms have even powers (or the constant term), the function is even; if all terms have odd powers, the function is odd.

Explore whether the function f(x) = -1 + 6x^6 - 9x^2 is even, odd, or neither. Understand the criteria for identifying even and odd functions, with step-by-step analysis and examples. Suitable for advanced high school students and beyond.

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