Question

1. (10 points) graph the function:

\\ f(x) = \

$$\begin{cases} -2 & \\text{for } x < 2 \\\\ -5 & \\text{for } x = 2 \\\\ 0 & \\text{for } x > 2 \\end{cases}$$

\\

Explanation:

Step1: Analyze the first piece

For \(x < 2\), \(f(x) = -2\), representing a horizontal line with an open circle at \((2, -2)\).

$$f(x) = -2 \quad \text{for} \quad x < 2$$

Step2: Analyze the second piece

For \(x = 2\), \(f(x) = -5\), representing a single solid point at \((2, -5)\).

$$f(2) = -5$$

Step3: Analyze the third piece

For \(x > 2\), \(f(x) = 0\), representing a horizontal line with an open circle at \((2, 0)\).

$$f(x) = 0 \quad \text{for} \quad x > 2$$

Answer:

The graph of the piecewise function \(f(x)\) consists of:

1. A horizontal ray along the line \(y = -2\) extending to the left from \(x = 2\), with an open circle at \((2, -2)\).
2. A single solid point at \((2, -5)\).
3. A horizontal ray along the line \(y = 0\) (the x-axis) extending to the right from \(x = 2\), with an open circle at \((2, 0)\).