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The image shows a problem related to **piecewise functions**. Here's a breakdown of the given information:

You are asked to consider the piecewise function \( f(x) \), defined as:
\[
f(x) =
  \begin{cases} 
   x^2 & \text{if } x < 2 \\
   5 & \text{if } 2 \leq x \leq 3 \\
   2x + 1 & \text{if } x > 3
  \end{cases}
\]

The question asks for the **domain** of this function in interval notation.

### Step-by-step solution:

- For \( x < 2 \), \( f(x) = x^2 \). This applies to all real values of \( x \) less than 2.
- For \( 2 \leq x \leq 3 \), \( f(x) = 5 \). This covers the interval from 2 to 3, including both endpoints.
- For \( x > 3 \), \( f(x) = 2x + 1 \). This applies to all real values of \( x \) greater than 3.

### Combining these intervals:
- The domain includes all values of \( x \) for which the function is defined. In this case, the function is defined for all \( x < 2 \), \( 2 \leq x \leq 3 \), and \( x > 3 \). Thus, the domain is all real numbers, or \( (-\infty, \infty) \).

However, based on the options in the image:
- The correct answer for the domain in interval notation is **\( (-\infty, \infty) \)**.

Would you like more details or clarification on this topic?

Here are some related questions for further practice:
1. What is the range of the given piecewise function?
2. How do you determine the continuity of this piecewise function?
3. What would the graph of this piecewise function look like?
4. How would you find the values of \( f(2) \) and \( f(3) \)?
5. How would the domain change if the function included a denominator or a square root?

**Tip:** When solving piecewise functions, carefully evaluate the behavior of the function at each boundary to ensure continuity or discontinuity.

What is the domain of the piecewise function in interval notation?

Learn how to find the domain of a piecewise function defined by different expressions over specified intervals. This problem helps you understand how to combine intervals to determine the overall domain, suitable for high school students studying piecewise functions.

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