Question

express the function graphed on the axes below as a piecewise function.
answer attempt 1 out of 2
$f(x) = \

$$\begin{cases} \\square \\text{ for } \\square \\\\ \\square \\text{ for } \\square \\end{cases}$$

$

Explanation:

Step1: Find top line's equation

First, identify two points on the upper line: $(-6, 3)$ (closed dot) and $(-2, -2)$ (open dot). Calculate the slope:
$m_1 = \frac{-2 - 3}{-2 - (-6)} = \frac{-5}{4} = -\frac{5}{4}$
Use point-slope form $y - y_1 = m(x - x_1)$ with $(-6, 3)$:
$y - 3 = -\frac{5}{4}(x + 6)$
Simplify to slope-intercept form:
$y = -\frac{5}{4}x - \frac{30}{4} + 3 = -\frac{5}{4}x - \frac{15}{2} + \frac{6}{2} = -\frac{5}{4}x - \frac{9}{2}$
The domain for this line is $x < -2$ (since the right end is an open dot at $x=-2$).

Step2: Find bottom line's equation

Identify two points on the lower line: $(0, -5)$ and $(3, -2)$ (closed dot), with an open dot at $(0, -6)$. Calculate the slope:
$m_2 = \frac{-2 - (-5)}{3 - 0} = \frac{3}{3} = 1$
Use slope-intercept form $y = mx + b$, where $b=-5$ (y-intercept):
$y = x - 5$
The domain for this line is $x > 0$ (since the left end is an open dot at $x=0$).

Answer:

$$ f(x) =
LATEXBLOCK0
$$