https://file.aiquickdraw.com/mathbot/file/1725976616131zpq4yy4t.jpg

The image contains two piecewise functions, \( f(x) \) and \( g(x) \), defined as follows:

### Function \( f(x) \):
\[
f(x) = 
\begin{cases} 
x - 1, & x \leq 1 \\
6 - 2x, & 1 < x < 3 \\
x + 1, & x \geq 3
\end{cases}
\]

### Function \( g(x) \):
\[
g(x) = 
\begin{cases} 
2x, & x < 1 \\
x + 1, & x \geq 1
\end{cases}
\]

These piecewise functions change their expressions based on different intervals of \( x \). The function \( f(x) \) has three intervals, while \( g(x) \) has two intervals.

Would you like to analyze a specific property of these functions (like continuity, limits, or composition), or would you like to perform operations such as finding \( f(g(x)) \) or \( g(f(x)) \)?

### Relative Questions:
1. What are the points of discontinuity, if any, in \( f(x) \) and \( g(x) \)?
2. What is the derivative of each piece of \( f(x) \) and \( g(x) \)?
3. Can you find \( f(g(x)) \) and \( g(f(x)) \)?
4. What are the limits of \( f(x) \) and \( g(x) \) as \( x \to 1 \) from both sides?
5. Can you graph these functions for better visualization?

### Tip:
When working with piecewise functions, always check the transition points between different intervals to determine if the function is continuous or has any jumps or gaps.

What are the properties and operations on the piecewise functions f(x) and g(x) defined in the image?

Explore the properties and operations on the piecewise functions f(x) and g(x), including function composition, continuity, and limits. Suitable for high school students studying advanced algebra.

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