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To find the zeros of the piecewise function \(g(x)\), we need to determine the values of \(x\) for which \(g(x) = 0\).

The given function \(g(x)\) is defined as:

\[
g(x) = 
\begin{cases} 
2x - 1, & x < 0 \\
x^2 - 4, & x \geq 0 
\end{cases}
\]

### Finding Zeros for Each Case

1. **Case 1: \(x < 0\), \(g(x) = 2x - 1\)**

   Set \(g(x) = 0\):

   \[
   2x - 1 = 0
   \]

   Solve for \(x\):

   \[
   2x = 1 \implies x = \frac{1}{2}
   \]

   Since \(x = \frac{1}{2} > 0\), it does not fall within the domain \(x < 0\) for this part of the piecewise function. Therefore, there are **no zeros** in the interval \(x < 0\).

2. **Case 2: \(x \geq 0\), \(g(x) = x^2 - 4\)**

   Set \(g(x) = 0\):

   \[
   x^2 - 4 = 0
   \]

   Solve for \(x\):

   \[
   x^2 = 4 \implies x = \pm 2
   \]

   Since we are considering the domain \(x \geq 0\), we only take the non-negative solution:

   \[
   x = 2
   \]

### Conclusion

The zero of the function \(g(x)\) is:
- \(x = 2\)

This is the only zero of the function within the specified domains of the piecewise parts.

Would you like any additional details or have any questions?

### Further Questions to Explore:
1. What are the zeros of \(g(x)\) if the function was defined for all real numbers?
2. How would you find the derivative of this piecewise function \(g(x)\)?
3. Can you find where the function \(g(x)\) is increasing or decreasing?
4. What is the range of the function \(g(x)\)?
5. How would you graph the piecewise function \(g(x)\) and identify its critical points?

#### Tip:
When dealing with piecewise functions, always check the conditions for each piece carefully to avoid errors in determining solutions, such as zeros or points of intersection.

Explore how to find the zeros of the piecewise function g(x) with a detailed step-by-step solution. This problem covers the concept of piecewise functions and involves solving for zeros in different domains. Suitable for high school students.

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