Question

write a piecewise function for the given graph. the piecewise function for the given graph is f(x) = \

$$\begin{cases} \\square & \\text{if } x \\geq \\square \\\\ \\square & \\text{if } x < \\square \\end{cases}$$

(simplify your answers.)

Explanation:

Step1: Find break point of the graph

The graph has a break at $x=-2$, so the piecewise function splits here: one part for $x\geq -2$, one for $x < -2$.

Step2: Find function for $x\geq -2$

Use points $(-5,5)$ and $(5,2)$. First calculate slope:
$m=\frac{2-5}{5-(-5)}=\frac{-3}{10}$
Use point-slope form $y-y_1=m(x-x_1)$ with $(-5,5)$:
$y-5 = \frac{-3}{10}(x+5)$
Simplify: $y = \frac{-3}{10}x - \frac{3}{2} + 5 = \frac{-3}{10}x + \frac{7}{2}$

Step3: Find function for $x < -2$

Use points $(-2,-5)$ and $(5,2)$. First calculate slope:
$m=\frac{2-(-5)}{5-(-2)}=\frac{7}{7}=1$
Use point-slope form $y-y_1=m(x-x_1)$ with $(-2,-5)$:
$y+5 = 1(x+2)$
Simplify: $y = x + 2 - 5 = x - 3$

Answer:

$$f(x)=
LATEXBLOCK0
$$