Question

Consider the following piece-wise function. Which of the below correctly describes the graph shown?

Answer 1

Answer:

C.

$f(x)=\left\{\begin{array}{l}\dfrac{1}{2}x-3,\ \ x$

Step-by-step explanation:

The graph of the piecewise function consists of two linear functions which graphs meets at point with x-coordinate x = 2.

1. The left linear function is determined for all x < 2 and is passing through the points (0,-3) and (-2,-4). So, its slope is

$\dfrac{-4-(-3)}{-2-0}=\dfrac{-1}{-2}=\dfrac{1}{2}$

and its equation is

$y-(-3)=\dfrac{1}{2}(x-0)\\ \\y=\dfrac{1}{2}x-3$

2. The right linear function is determined for all x ≥ 2 and is passing through the points (2,-2) and (3,1). So, its slope is

$\dfrac{1-(-2)}{3-2}=\dfrac{3}{1}=3$

and its equation is

$y-1=3(x-3)\\ \\y=3x-8$

3. Hence, the expression for the piece-wise function is

$f(x)=\left\{\begin{array}{l}\dfrac{1}{2}x-3,\ \ x$