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The function \( g(x) \) is given as a piecewise function:

\[
g(x) =
\begin{cases} 
x^2 - 6x + 13 & \text{if } x \geq 1 \\
2\sqrt{2 - x} & \text{if } x < 1
\end{cases}
\]

To evaluate \( g(1) \), we use the first piece of the piecewise function, since \( g(x) = x^2 - 6x + 13 \) for \( x \geq 1 \).

Substituting \( x = 1 \):

\[
g(1) = 1^2 - 6(1) + 13 = 1 - 6 + 13 = 8
\]

Thus, \( g(1) = 8 \).

Would you like any more details on this, or have other questions?

Here are five related questions to expand on this concept:

1. How do you evaluate a derivative using the limit definition for piecewise functions?
2. How do you check if a piecewise function is continuous at a given point?
3. What does the existence of a tangent line at a point tell you about the function's behavior?
4. How do one-sided limits help in determining the differentiability of a function?
5. Can you describe a situation where a function has a jump discontinuity?

**Tip:** When dealing with piecewise functions, always ensure you're using the correct "piece" of the function for the interval you're evaluating.

Evaluate g(1) in part g) ii) of the given problem.

Learn how to evaluate g(1) for a piecewise function where g(x) is defined by two different expressions for x ≥ 1 and x < 1. This problem is ideal for Calculus I students, involving the use of basic function evaluation and continuity principles.

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