https://file.aiquickdraw.com/mathbot/file/17276534200172ndcmhdi.jpg

The image shows three piecewise functions. Let's break them down:

### (i) 
\[
f(x) = 
\begin{cases} 
x + 2 & \text{if } x \leq 1 \\
3 + 5 \ln(x) & \text{if } x > 1 
\end{cases}
\]

### (ii) 
\[
f(x) = 
\begin{cases} 
7e^x - 4 & \text{if } x < 0 \\
3 \cos(x) & \text{if } x \geq 0 
\end{cases}
\]

### (iii)
\[
f(x) = 
\begin{cases} 
\frac{x^2 - 49}{x - 7} & \text{if } x \neq 7 \\
12 & \text{if } x = 7 
\end{cases}
\]

Would you like me to evaluate or analyze any of these functions in more detail (e.g., continuity, differentiability, or solving for specific values)?

Here are 5 related questions to expand on the topic:
1. What is the continuity of function \( f(x) \) for each case?
2. Can you compute the limit of \( f(x) \) as \( x \to 1 \) in part (i)?
3. What is the value of \( f(x) \) when \( x = 7 \) in part (iii)?
4. How do you determine if a function is differentiable at a point of discontinuity?
5. Can you explain the behavior of \( f(x) \) in part (ii) as \( x \to 0 \) from both sides?

**Tip:** For piecewise functions, always check for continuity and differentiability at the transition points where the function definition changes.

What are the piecewise functions f(x) given in the image, and how do they behave?

Explore piecewise functions with detailed analysis of continuity, limits, and differentiability. This example includes logarithmic, exponential, and trigonometric functions, suitable for students in Grades 11-12. Learn how to check for function behavior at critical points.

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