 f_1 = \alpha \left( \frac{(i_0 - i_1)^2}{(c_0 - i_1)^2} - \frac{(i_0 + i_1)^2}{(c_0 + i_1)^2} \right) 
f_2 = \alpha \left( \frac{(i_0 - i_2)^2}{(c_0 - i_2)^2} + \frac{(i_0 + i_2)^2}{(c_0 + i_2)^2} \right)  

You provided two functions:

\[
f_1 = \alpha \left( \frac{(i_0 - i_1)^2}{(c_0 - i_1)^2} - \frac{(i_0 + i_1)^2}{(c_0 + i_1)^2} \right)
\]
\[
f_2 = \alpha \left( \frac{(i_0 - i_2)^2}{(c_0 - i_2)^2} + \frac{(i_0 + i_2)^2}{(c_0 + i_2)^2} \right)
\]

### Key points:
- Both functions contain the constant \(\alpha\).
- The expressions involve terms with \(i_0\), \(i_1\), \(i_2\), and \(c_0\), which could be interpreted as either current (if referring to electrical terms) or some other variables depending on context.

### Differences:
1. \(f_1\) uses a **difference** between two rational expressions involving \((i_0 - i_1)\) and \((i_0 + i_1)\).
2. \(f_2\) uses a **sum** of two rational expressions involving \((i_0 - i_2)\) and \((i_0 + i_2)\).

Would you like to explore specific aspects of these equations, such as their behavior under certain conditions, differentiation, or simplification?

### Potential questions to deepen your understanding:
1. How does changing \(\alpha\) impact the values of \(f_1\) and \(f_2\)?
2. What happens to \(f_1\) and \(f_2\) if \(i_1 = i_2\)?
3. How do \(f_1\) and \(f_2\) behave as \(i_1\) or \(i_2\) approach \(c_0\)?
4. Can \(f_1\) or \(f_2\) ever be zero? Under what conditions?
5. How do these expressions change if \(i_0 = 0\)?

**Tip:** When dealing with complex expressions, checking for symmetry or certain limiting cases can help simplify the analysis.

f_1 = \alpha \left( \frac{(i_0 - i_1)^2}{(c_0 - i_1)^2} - \frac{(i_0 + i_1)^2}{(c_0 + i_1)^2} \right) 
f_2 = \alpha \left( \frac{(i_0 - i_2)^2}{(c_0 - i_2)^2} + \frac{(i_0 + i_2)^2}{(c_0 + i_2)^2} \right)

This problem explores two complex functions involving rational expressions with variables i_0, i_1, i_2, and c_0, and the constant alpha. Delve into differences and sums of rational terms and their impact on function behavior, suitable for undergraduate-level mathematics.

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