https://file.aiquickdraw.com/mathbot/file/17256053121623x58w99l.jpg

The image contains a piecewise function \( f(x) \) defined as follows:

\[
f(x) = 
\begin{cases} 
gx + b & \text{for } x \leq -1 \\
x - 3 & \text{for } -1 < x \leq 1 \\
2x + 4 & \text{for } x > 1
\end{cases}
\]

If you'd like me to solve or explore a specific question regarding this function, feel free to ask!

Here are five potential questions to expand on this:

1. How can we determine the values of \( g \) and \( b \) if we want the function to be continuous?
2. What is the derivative of \( f(x) \) for each piece of the piecewise function?
3. How do you find the limits of \( f(x) \) as \( x \) approaches \( -1 \) and \( 1 \) from both sides?
4. Can the piecewise function be integrated over a given interval?
5. How do you determine whether \( f(x) \) is differentiable at the points where the function definition changes?

**Tip:** Continuity at the points where the function changes can be ensured by equating the values of the function at those points from both sides.

Find the piecewise function and related properties from the given image.

Explore the piecewise function f(x) defined in three parts with conditions. Analyze its continuity, differentiation, and limits at key points such as x = -1 and x = 1. Suitable for students in Grades 11-12.

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