Question

1. graph the following piecewise function. (4pts)

$f(x) \

$$\begin{cases} 3x - 1, & x \\geq -1 \\\\ -5, & x < -1 \\end{cases}$$

$

Explanation:

Step1: Analyze the first piece ($x \geq -1$)

The function is $f(x) = 3x - 1$ for $x \geq -1$. This is a linear function with slope $3$ and y-intercept $-1$. At $x = -1$, $f(-1) = 3(-1) - 1 = -4$? Wait, no, wait: $3(-1) - 1 = -3 -1 = -4$? Wait, but let's check the graph. Wait, maybe I made a mistake. Wait, when $x = -1$, $f(-1) = 3(-1) -1 = -4$? But the graph shown has a point at $x=-1$ with $y$ around... Wait, maybe the original graph has an error, but let's proceed. For $x \geq -1$, we can plot points. Let's take $x = -1$: $f(-1) = 3(-1) -1 = -4$? Wait, no, wait the function is $3x -1$. So when $x = 0$, $f(0) = -1$; $x = 1$, $f(1) = 2$; $x = 2$, $f(2) = 5$, etc. So this is a line with slope 3, passing through $(0, -1)$, $(1, 2)$, etc., starting at $x = -1$ (with a closed dot at $x=-1$ because $x \geq -1$).

Step2: Analyze the second piece ($x < -1$)

The function is $f(x) = -5$ for $x < -1$. This is a horizontal line at $y = -5$ for all $x$ values less than $-1$. So we need to draw a horizontal line at $y = -5$ for $x < -1$, with an open dot at $x = -1$ (since $x < -1$ does not include $x=-1$).

Step3: Correct the graph (if needed)

Looking at the given graph, the red line seems to have a slope of 3, but at $x=-1$, the point should be $f(-1) = 3(-1) -1 = -4$, but the graph shows a different point. However, to graph correctly:

• For $x < -1$: Draw a horizontal line at $y = -5$ (so a horizontal line left of $x=-1$).
• For $x \geq -1$: Draw the line $y = 3x -1$, starting at $x=-1$ (closed dot) and going upwards with slope 3.

Answer:

To graph the piecewise function:

1. For \( x < -1 \): Draw a horizontal line at \( y = -5 \) (open dot at \( x = -1 \) since \( x < -1 \) does not include \( x = -1 \)).
2. For \( x \geq -1 \): Draw the line \( y = 3x - 1 \) (closed dot at \( x = -1 \) since \( x \geq -1 \) includes \( x = -1 \)), with slope \( 3 \) (passing through points like \( (-1, -4) \), \( (0, -1) \), \( (1, 2) \), etc.).

(Note: The provided graph may have an error in the \( y \)-value at \( x = -1 \) for the linear piece, but the correct graph follows the above steps.)