https://file.aiquickdraw.com/mathbot/file/1729708499401a1nc61wr.jpg

The piecewise function shown in the image is:

\[
f(x) = \begin{cases} 
2x + 3, & \text{if } x \leq 0 \\
\frac{1}{2}x - 1, & \text{if } x > 0 
\end{cases}
\]

Let's analyze each part of this function:

1. **For \(x \leq 0\)**: The function is \(f(x) = 2x + 3\), which is a linear equation with slope 2 and y-intercept 3. This part of the graph will be a line with a positive slope, starting at \(x = 0\), where \(f(0) = 3\), and continuing downwards as \(x\) becomes more negative.

2. **For \(x > 0\)**: The function is \(f(x) = \frac{1}{2}x - 1\), which is also a linear equation with slope \( \frac{1}{2} \) and y-intercept \(-1\). This part of the graph will be a line with a shallower positive slope, starting from \(x = 0^+\) (just greater than zero) and extending rightwards.

To understand the graph better, let's calculate a few key points:

### For \(x \leq 0\) (\(f(x) = 2x + 3\)):
- At \(x = 0\): \(f(0) = 2(0) + 3 = 3\)
- At \(x = -1\): \(f(-1) = 2(-1) + 3 = 1\)
- At \(x = -2\): \(f(-2) = 2(-2) + 3 = -1\)

### For \(x > 0\) (\(f(x) = \frac{1}{2}x - 1\)):
- At \(x = 0^+\) (right after zero): \(f(0^+) = \frac{1}{2}(0) - 1 = -1\)
- At \(x = 2\): \(f(2) = \frac{1}{2}(2) - 1 = 0\)
- At \(x = 4\): \(f(4) = \frac{1}{2}(4) - 1 = 1\)

### Key Observation:
At \(x = 0\), the two pieces of the function meet at different points:
- For \(x \leq 0\), the function equals 3 at \(x = 0\).
- For \(x > 0\), the function equals \(-1\) at \(x = 0^+\).
So, there is a **jump discontinuity** at \(x = 0\) because the left and right limits of the function do not match.

This piecewise function will have two lines, but with a break or jump at \(x = 0\).

Would you like further details on graphing or any other clarification?

Here are some related questions to deepen your understanding:
1. How do we calculate the slope of a linear equation?
2. What is a jump discontinuity in a piecewise function?
3. How do you find the y-intercept of a linear equation?
4. How do you sketch a piecewise function on a coordinate plane?
5. How do limits relate to continuity in a piecewise function?

**Tip:** When sketching piecewise functions, carefully check where each piece starts and whether the pieces connect smoothly or with a gap.

Which graph is the graph of the piecewise function: f(x) = {2x + 3, x ≤ 0; (1/2)x - 1, x > 0}?

Explore how to graph the piecewise function f(x) = {2x + 3 for x ≤ 0, (1/2)x - 1 for x > 0}. This solution includes an analysis of linear equations, calculations of key points, and a discussion of the jump discontinuity at x = 0. Suitable for high school students in Grades 9-12.

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