https://file.aiquickdraw.com/mathbot/file/173438973690478uwljgg.jpg

The image shows a **piecewise-defined function** \( f(x) \), given by:

\[
f(x) =
\begin{cases}
x^{-1}, & x < 0 \\
-x + 1, & 0 < x \leq 1 \\
\log x, & x \geq 1
\end{cases}
\]

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### Explanation of Each Case:
1. **For \( x < 0 \):**  
   The function is defined as \( f(x) = x^{-1} \), or \( \frac{1}{x} \).  
   This means the reciprocal of \( x \) for all negative values of \( x \).

2. **For \( 0 < x \leq 1 \):**  
   The function is defined as \( f(x) = -x + 1 \).  
   This is a linear equation with a negative slope (-1) and a y-intercept of 1.

3. **For \( x \geq 1 \):**  
   The function is \( f(x) = \log x \), which is the natural logarithm of \( x \) (base \( e \)).  
   This is valid for \( x \geq 1 \) because the logarithmic function is only defined for positive \( x \).

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Let me know if you'd like me to analyze this function further, such as:  
- Continuity or differentiability at the boundaries \( x = 0 \) and \( x = 1 \).  
- Sketching the graph of \( f(x) \).  
- Domain and range analysis.  

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### Related Questions:
1. Is the function \( f(x) \) continuous at \( x = 0 \) and \( x = 1 \)?  
2. What are the derivatives of \( f(x) \) for each piece?  
3. How does \( f(x) \) behave as \( x \to -\infty \) and \( x \to \infty \)?  
4. What is the domain and range of \( f(x) \)?  
5. Can you graph the piecewise function \( f(x) \)?

### Tip:  
When analyzing a piecewise function, always check the transition points (boundaries) for continuity and differentiability.

The function f(x) is defined piecewise as follows:
- f(x) = x^{-1}, x < 0
- f(x) = -x + 1, 0 < x <= 1
- f(x) = log x, x >= 1

Learn how to analyze a piecewise-defined function f(x), consisting of a reciprocal function x^{-1}, a linear equation -x + 1, and a logarithmic function log x. Explore continuity, differentiability, and behavior at the boundaries x = 0 and x = 1. Suitable for Grades 11-12.

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