Explanation:
To determine whether the function \(f(x) = \sqrt[3]{x}\sin(x)/x^4 - 7|x| - \tan^2(x)\) is even, odd, or neither, we can analyze its symmetry properties:
1. **Even Function:** A function \(f(x)\) is even if \(f(-x) = f(x)\) for all \(x\) in its domain. In other words, it's symmetric about the y-axis.
2. **Odd Function:** A function \(f(x)\) is odd if \(f(-x) = -f(x)\) for all \(x\) in its domain. In other words, it's symmetric about the origin.
Let's examine each term of the function individually:
- The first term, \(\sqrt[3]{x}\sin(x)/x^4\), is neither even nor odd because it doesn't satisfy the properties for either symmetry.
- The second term, \(7|x|\), is an even function because \(|x|\) is even, meaning it's symmetric about the y-axis.
- The third term, \(\tan^2(x)\), is neither even nor odd because it doesn't satisfy the properties for either symmetry.
Since the function is a combination of these terms (with the first term not having even or odd symmetry), the overall function \(f(x)\) is neither even nor odd.
Hopefully i helped u out! :)