𝑓(𝑥) = {
−𝑥2 + 8, 𝑖𝑓 𝑥 < 0
−3𝑥 + 7, 𝑖𝑓 0 ≤ 𝑥 < 4
√𝑥 − 4 + 1, 𝑖𝑓 4 ≤ 𝑥

The function you've provided is a piecewise function, which means that it is defined by different expressions depending on the value of \( x \). Let's break it down:

\[
f(x) = 
\begin{cases} 
-x^2 + 8, & \text{if } x < 0 \\
-3x + 7, & \text{if } 0 \leq x < 4 \\
\sqrt{x - 4} + 1, & \text{if } 4 \leq x 
\end{cases}
\]

This function is defined as:

1. **For \( x < 0 \)**: The function follows the equation \( f(x) = -x^2 + 8 \), which is a downward-opening parabola.
2. **For \( 0 \leq x < 4 \)**: The function follows the equation \( f(x) = -3x + 7 \), which is a straight line with slope -3 and y-intercept 7.
3. **For \( 4 \leq x \)**: The function follows the equation \( f(x) = \sqrt{x - 4} + 1 \), which represents a square root function shifted 4 units to the right and 1 unit upward.

If you need help evaluating the function for a specific \( x \)-value or want to explore its behavior at key points, feel free to ask!

Would you like to see a graph of this function, or do you need any other specific details? 

### Questions for further exploration:
1. How would the function behave at \( x = 0 \) and \( x = 4 \)?
2. Can you explain how to evaluate \( f(x) \) at a given point in each of the three intervals?
3. What are the potential limits of \( f(x) \) as \( x \to -\infty \) and \( x \to \infty \)?
4. What is the derivative of each piece of the function, and how does it change across the intervals?
5. How do you determine the continuity of this piecewise function at the boundaries (i.e., at \( x = 0 \) and \( x = 4 \))?

**Tip**: Always check the function values at the boundaries of each piecewise interval to ensure proper continuity and smooth transitions.

f(x) = {−x^2 + 8, if x < 0; −3x + 7, if 0 ≤ x < 4; sqrt(x − 4) + 1, if 4 ≤ x}

Explore the properties of a piecewise function defined by quadratic, linear, and square root expressions. Learn how to evaluate the function at different intervals, determine continuity, and analyze key points. Suitable for high school students (Grades 9-12).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation