To determine whether the function $f(x) = -\frac{5}{x^2} + 9x^6$ is even, odd, or neither, we need to analyze its behavior under the transformation $x \to -x$.

Step 1: Test for an even function

A function is even if $f(-x) = f(x)$ for all $x$ in the domain of the function.

Let's compute $f(-x)$:

$f(-x) = -\frac{5}{(-x)^2} + 9(-x)^6$

• The term $(-x)^2 = x^2$, so $\frac{5}{(-x)^2} = \frac{5}{x^2}$.
• The term $(-x)^6 = x^6$, because raising any negative number to an even power results in a positive number.

Thus,

$f(-x) = -\frac{5}{x^2} + 9x^6$

This is exactly the same as the original function $f(x) = -\frac{5}{x^2} + 9x^6$. Therefore, $f(x)$ is an even function.

Step 2: Test for an odd function

A function is odd if $f(-x) = -f(x)$ for all $x$ in the domain of the function.

For $f(x)$, we have shown that $f(-x) = f(x)$, so the function cannot be odd.

Conclusion:

The function $f(x) = -\frac{5}{x^2} + 9x^6$ is even.

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Would you like a more detailed explanation or further questions on this topic? Here are some related questions to explore:

1. How can we determine if a rational function is even or odd?
2. What are some examples of odd functions?
3. How do the powers of $x$ in a polynomial affect whether the function is even or odd?
4. Can a function be both even and odd?
5. What is the graphical interpretation of even and odd functions?

Tip: When checking for even and odd functions, always replace $x$ with $-x$ and compare the result with the original function.