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The problem asks to find the values of \(a\) and \(b\) so that the function \( f(x) \) is continuous at \( x = 1 \).

The given function is:

\[
f(x) = 
\begin{cases} 
a - x, & \text{if } x \leq 1, \\
2x - b, & \text{if } x > 1.
\end{cases}
\]

For the function to be continuous at \(x = 1\), the left-hand limit (as \(x\) approaches 1 from the left) must equal the right-hand limit (as \(x\) approaches 1 from the right) and must also equal the function value at \(x = 1\).

### Steps to find \(a\) and \(b\):
1. **Left-hand limit** (\(x \to 1^-\)):
   \[
   \lim_{x \to 1^-} f(x) = a - 1.
   \]
   
2. **Right-hand limit** (\(x \to 1^+\)):
   \[
   \lim_{x \to 1^+} f(x) = 2(1) - b = 2 - b.
   \]
   
3. **Continuity condition at \(x = 1\)**:
   The function is continuous at \(x = 1\) if the limits from both sides are equal:
   \[
   a - 1 = 2 - b.
   \]

### Solving the equation:
\[
a - 1 = 2 - b.
\]
Rearrange the equation:
\[
a + b = 3.
\]

Thus, \(a\) and \(b\) must satisfy the equation \(a + b = 3\).

Let me know if you'd like more details on solving or verifying this, or any other questions!

### Related questions:
1. What is the definition of continuity at a point in terms of limits?
2. How can we check if a piecewise function is continuous at a given point?
3. How does the behavior of a function differ at points where it's not continuous?
4. Can the function in this problem be made differentiable at \(x = 1\) by choosing specific values of \(a\) and \(b\)?
5. How would the solution change if the function were defined on different intervals?

### Tip:
For piecewise functions, always check continuity by ensuring the function values from both sides of the critical point are equal!

Tentukan a dan b agar fungsi berikut kontinu di x = 1. f(x) = { a - x, x <= 1; 2x - b, x > 1 }

This example demonstrates how to find the values of a and b so that a piecewise function is continuous at x = 1. The solution involves checking limits from both sides of the critical point and solving the resulting equation. Suitable for high school or early college students.

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