Question

express the function graphed on the axes below as a piecewise function.

Explanation:

Step1: Identify left segment points

Points: $(1, -4)$ (open), $(4, 2)$ (closed)

Step2: Find left segment slope

Slope $m=\frac{2-(-4)}{4-1}=\frac{6}{3}=2$

Step3: Find left segment equation

Use point-slope form: $y - (-4)=2(x - 1)$
Simplify: $y=2x - 6$, domain $1 < x < 4$

Step4: Identify right segment points

Points: $(4, 3)$ (open), $(7, 0)$

Step5: Find right segment slope

Slope $m=\frac{0-3}{7-4}=\frac{-3}{3}=-1$

Step6: Find right segment equation

Use point-slope form: $y - 3=-1(x - 4)$
Simplify: $y=-x + 7$, domain $x \geq 4$

Step7: Combine into piecewise function

Assemble the two cases with domains

Answer:

$$ f(x)=
LATEXBLOCK0
$$