Suppose that the function is defined for all real numbers as follows f(x) = 6 if x < - 4; - 10 + x ^ 2 if - 4 <= x < 4; 2x - 2 if x >= 4 Graph the function f. Then determine whether or not the function is continuous .

### Step 1: Analyze the Piecewise Function

The function \( f(x) \) is defined as:

\[
f(x) = 
\begin{cases} 
6 & \text{if } x < -4 \\
-10 + x^2 & \text{if } -4 \leq x < 4 \\
2x - 2 & \text{if } x \geq 4 
\end{cases}
\]

### Step 2: Graph the Function

To graph the function, we'll plot each piece of the function separately:

1. **For \( x < -4 \)**: The function is constant at \( f(x) = 6 \). This is a horizontal line at \( y = 6 \) extending leftward from \( x = -4 \).
   
2. **For \( -4 \leq x < 4 \)**: The function is a quadratic expression \( f(x) = -10 + x^2 \). This represents a parabola that opens upwards with its vertex shifted down by 10 units from the standard parabola \( y = x^2 \).
   
3. **For \( x \geq 4 \)**: The function is a linear expression \( f(x) = 2x - 2 \). This represents a line with a slope of 2, starting at \( x = 4 \).

### Step 3: Check Continuity at the Transition Points

A function is continuous at a point if the following three conditions are met:
1. The function is defined at the point.
2. The limit of the function as it approaches the point from both sides exists.
3. The limit equals the function value at that point.

#### **Check at \( x = -4 \)**:
- **Left-hand limit**: As \( x \) approaches \(-4\) from the left, \( f(x) = 6 \).
- **Right-hand limit**: As \( x \) approaches \(-4\) from the right, \( f(x) = -10 + (-4)^2 = -10 + 16 = 6 \).
- **Function value**: \( f(-4) = -10 + (-4)^2 = 6 \).

Since the left-hand limit, right-hand limit, and function value at \( x = -4 \) are all equal to 6, the function is continuous at \( x = -4 \).

#### **Check at \( x = 4 \)**:
- **Left-hand limit**: As \( x \) approaches \( 4 \) from the left, \( f(x) = -10 + 4^2 = 6 \).
- **Right-hand limit**: As \( x \) approaches \( 4 \) from the right, \( f(x) = 2(4) - 2 = 8 - 2 = 6 \).
- **Function value**: \( f(4) = 2(4) - 2 = 6 \).

Since the left-hand limit, right-hand limit, and function value at \( x = 4 \) are all equal to 6, the function is continuous at \( x = 4 \).

### Conclusion

Since the function is continuous at all transition points \( x = -4 \) and \( x = 4 \), and all other parts are continuous by themselves, the entire function \( f(x) \) is **continuous for all real numbers**.

Would you like a detailed sketch of the graph? Or do you have any questions? Here are some related problems to consider:

1. What if the function had a discontinuity at one of the transition points?
2. How would the continuity change if the function had a different form for one of its parts?
3. How does the continuity of a piecewise function relate to the limits?
4. What would the graph look like if the quadratic part was shifted or stretched?
5. How does a piecewise function behave if one of the conditions changes to strict inequality?

**Tip**: Always check both left-hand and right-hand limits when verifying the continuity of a piecewise function.

Explore the graph and continuity analysis of the piecewise function f(x) = 6 for x < -4, -10 + x^2 for -4 ≤ x < 4, and 2x - 2 for x ≥ 4. Understand the conditions for continuity at transition points and the overall function. Suitable for advanced high school students.

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