f(x)={ 
2x−2
−(x−3) 
2
 +6
​
  
 for x≤−1 
1
1
​
 
 for x≥3 
1
1
​
 
​
 
Find , f, of, 0
Find f(0)

The function \( f(x) \) is defined piecewise, meaning that different formulas are used depending on the value of \( x \).

Let's analyze the given function.

### Definition of the function:
- \( f(x) = 2x - 2 - (x - 3)^2 + 6 \) for \( x \leq -1 \)
- \( f(x) = 11 \) for \( x \geq 3 \)

We need to determine which part of the piecewise function to use for \( f(0) \).

Since \( 0 \) is not included in the intervals \( x \leq -1 \) or \( x \geq 3 \), the function does not provide a definition for \( x = 0 \). Therefore, \( f(0) \) is **undefined** based on the given information.

If you'd like more clarification or further assistance with a different approach to this problem, feel free to ask.

### Relative questions:
1. How do you determine the domain of a piecewise function?
2. What are the steps to evaluate a piecewise function at a specific point?
3. Can the function be redefined to include \( x = 0 \)? If so, how?
4. How would \( f(0) \) change if the intervals for \( f(x) \) were different?
5. How can you graph a piecewise function based on its definition?

### Tip:
When working with piecewise functions, always check which condition the input value satisfies before selecting the corresponding expression.

f(x) = { 2x−2−(x−3)^2 +6 for x ≤ -1, 11 for x ≥ 3 }. Find f(0).

Understand how to evaluate a piecewise function and determine the value of f(0) when it's undefined within given conditions. This guide covers function evaluation, domain analysis, and piecewise function definitions, suitable for high school students in Grades 10-12.

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