https://file.aiquickdraw.com/mathbot/file/17368748349598b87bqug.jpg

The piecewise function provided is:

\[
f(x) = \begin{cases} 
x & \text{for } x < -1 \text{ and } x > 1 \\
x^2 + ax + b & \text{for } -1 \leq x \leq 1 
\end{cases}
\]

This function is defined differently for different intervals of \(x\). Let's explore its properties, especially continuity and differentiability, which often require matching conditions at the boundaries (\(x = -1\) and \(x = 1\)).

### Continuity Check:
To ensure continuity at \(x = -1\) and \(x = 1\), the function values from both intervals must match.

1. **At \(x = -1\)**:
   - From \(x < -1\): \(f(-1) = -1\)
   - From \(-1 \leq x \leq 1\): \(f(-1) = (-1)^2 + a(-1) + b = 1 - a + b\)

   For continuity at \(x = -1\),
   \[
   -1 = 1 - a + b \implies a - b = 2
   \]

2. **At \(x = 1\)**:
   - From \(x > 1\): \(f(1) = 1\)
   - From \(-1 \leq x \leq 1\): \(f(1) = 1^2 + a(1) + b = 1 + a + b\)

   For continuity at \(x = 1\),
   \[
   1 = 1 + a + b \implies a + b = 0
   \]

### Solving for \(a\) and \(b\):
From the two equations:
1. \(a - b = 2\)
2. \(a + b = 0\)

Adding and subtracting these equations:
- \(2a = 2 \implies a = 1\)
- Substituting \(a = 1\) into \(a + b = 0\): \(1 + b = 0 \implies b = -1\)

Thus, \(a = 1\) and \(b = -1\).

### Final Function:
\[
f(x) = \begin{cases} 
x & \text{for } x < -1 \text{ and } x > 1 \\
x^2 + x - 1 & \text{for } -1 \leq x \leq 1 
\end{cases}
\]

Would you like a detailed explanation of the steps or have any questions?

Here are 5 related questions for further exploration:
1. How do we check if the function is differentiable at \(x = -1\) and \(x = 1\)?
2. What is the graphical interpretation of a piecewise function's continuity and differentiability?
3. How can we generalize the process of finding constants \(a\) and \(b\) for other piecewise functions?
4. How would the solution change if the intervals of the piecewise function were different?
5. Can we explore the second derivative for the interval \(-1 \leq x \leq 1\) and its implications?

**Tip**: Always ensure the boundary points of a piecewise function are evaluated carefully for both continuity and differentiability.

Determine the constants a and b such that the piecewise function f(x) is continuous for all x.

Learn how to find constants a and b to ensure continuity of a piecewise function. This solution explores the conditions for continuity at boundary points and solves a system of equations step-by-step. Perfect for high school students studying advanced algebra.

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