To determine if a function is even, odd, or neither, we need to use the following definitions:

• Even function: $f(-x) = f(x)$
• Odd function: $f(-x) = -f(x)$
• Neither: If neither of the above properties holds.

Let’s analyze each function given in the question:

1. $e^x$:

   • $f(x) = e^x$
   • $f(-x) = e^{-x}$, which is not equal to $e^x$ (so it's not even) and not equal to $-e^x$ (so it's not odd).
   • Conclusion: Neither even nor odd.

2. $e^{-|x|}$:

   • Since $|x|$ is always non-negative, $f(x) = e^{-|x|}$.
   • For $f(-x)$, we get $e^{-|x|}$, which is the same as $f(x)$.
   • Conclusion: Even function.

3. $x^3 \cos(nx)$:

   • $f(x) = x^3 \cos(nx)$
   • For $f(-x)$, we get $(-x)^3 \cos(-nx) = -x^3 \cos(nx) = -f(x)$.
   • Conclusion: Odd function.

4. $x^2 \tan(\pi x)$:

   • $f(x) = x^2 \tan(\pi x)$
   • For $f(-x)$, we get $(-x)^2 \tan(-\pi x) = x^2 (-\tan(\pi x)) = -x^2 \tan(\pi x)$, which is not equal to $f(x)$ or $-f(x)$.
   • Conclusion: Neither even nor odd.

5. $\sinh(x) - \cosh(x)$:

   • We use the definitions $\sinh(x) = \frac{e^x - e^{-x}}{2}$ and $\cosh(x) = \frac{e^x + e^{-x}}{2}$.
   • So, $f(x) = \sinh(x) - \cosh(x) = \frac{e^x - e^{-x}}{2} - \frac{e^x + e^{-x}}{2} = -e^{-x}$.
   • For $f(-x) = -e^{x}$, which is not equal to $f(x)$ and not equal to $-f(x)$.
   • Conclusion: Neither even nor odd.

Summary:

1. $e^x$: Neither even nor odd.
2. $e^{-|x|}$: Even.
3. $x^3 \cos(nx)$: Odd.
4. $x^2 \tan(\pi x)$: Neither even nor odd.
5. $\sinh(x) - \cosh(x)$: Neither even nor odd.

Would you like details on any of these evaluations or further explanations?

Here are 5 questions that extend this discussion:

1. How do we determine if a trigonometric function alone is even or odd?
2. What are some applications of even and odd functions in Fourier series?
3. How do absolute values affect the symmetry of functions?
4. Why is the function $e^x$ neither even nor odd, despite its simple form?
5. Can composite functions be analyzed for evenness and oddness in a similar way?

Tip: Remember that an even function is symmetric around the y-axis, while an odd function is symmetric around the origin.