find f(x) = x + 1 if x < 2
find f(x) = x² if x ≥ -1

The problem describes a piecewise function with two conditions. However, it seems there may be an overlap in the domains provided. To clarify, I'll address each part separately and note any possible conflicts.

### Given:
1. \( f(x) = x + 1 \) if \( x < 2 \)
2. \( f(x) = x^2 \) if \( x \geq -1 \)

### Analysis:

- The first condition \( f(x) = x + 1 \) is defined for \( x < 2 \).
- The second condition \( f(x) = x^2 \) is defined for \( x \geq -1 \).

### Conflict:
There's a domain overlap between \( -1 \leq x < 2 \). In this range, both conditions would apply. In such a case, the problem usually specifies which function to use in the overlapping region. If this is not specified, it creates ambiguity.

### Piecewise Function Representation:
Assuming the function \( f(x) = x + 1 \) is meant to apply only for \( x < -1 \) to avoid overlap, we can redefine the function as follows:

\[
f(x) =
\begin{cases}
x + 1 & \text{if } x < -1 \\
x^2 & \text{if } x \geq -1
\end{cases}
\]

This resolves the overlap issue.

### Conclusion:
If you intended to have the function as initially stated, there would be a need for a specific rule about which function to apply in the overlapping region.

Would you like further details or have any questions?

Here are some related questions:

1. What would happen if \( f(x) = x + 1 \) for \( x \leq 2 \) instead of \( x < 2 \)?
2. How would the piecewise function change if \( f(x) = x^2 \) were defined for \( x \geq 2 \)?
3. What if there were a third condition, \( f(x) = \sin(x) \) for \( x \) in a different interval?
4. How do you graph a piecewise function with overlapping conditions?
5. How would this function look if it were only one continuous function?

**Tip:** When dealing with piecewise functions, always check for domain overlaps to ensure the function is well-defined throughout its domain.

Explore the piecewise function f(x) = x + 1 if x < 2 and f(x) = x^2 if x ≥ -1. Learn how to resolve domain overlaps and clarify conditions in piecewise functions. Suitable for high school students studying advanced functions and domains.

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