Question

On a coordinate plane, a piecewise function has 2 lines. The first line has an open circle at (0, negative 2) and continues up through (negative 5, 3) with an arrow instead of an endpoint. The second line has a closed circle at (0, 0) and continues down with a negative slope through (4, negative 2) with an arrow instead of an endpoint. Which defines the piecewise function shown?

Answer 1

$\bold{\text{Answer:}\quad f(x)=\bigg\{\begin{array}{ll} -x-2&;x<0\\ -\frac{1}{2}x&;x\geq0 \end{array}}$

Step-by-step explanation:

Find the equation of each line in Slope-Intercept form: y = mx + b

Line 1 passes through (-5, 3) and (0, -2)   -->   m = -1,  b = -2

                                                                   -->  y = -x - 2

Line 2 passes through (0, 0) and (4, -2)   -->   m = -1/2,  b = 0

                                                                   -->   y = -(1/2)x + 0

Now that you have the equations, evaluate which x-values are included.

Line 1: open dot at x = 0 and line goes to the left        -->  x < 0

Line 2: closed dot at x = 0 and line goes to the right   --> x ≥ 0

Now you have all the information you need to write the piecewise function:

                                $\large\boxed{f(x)=\bigg\{\begin{array}{ll} -x-2&;x<0\\ -\frac{1}{2}x&;x\geq0 \end{array}}$

Answer 2

The piece wise function can be written as

$f(x) = -x-2 \; for\; x < 0$

$f(x) = \dfrac{-1}{2}x \; for\; x \geq 0$

The Equation of line passing through  $(x_1,y_1)\; and \; (x_2,y_2)$ can be given  by equation (1)

$y -y_1 = \dfrac{y_2 -y_1 }{x_2 - x_1} (x-x_1) ........(1)$

The equation of line passing through (0,-2) and (-5,3) =

$y-(-2) = \dfrac{3-(-2)}{-5-0}\times (x-0)$

$y +2 = -x\\y = -x -2 ......(2)$

Similarly the line passing through (0,0) and (4,-2) =

$y- 0 = \dfrac{-2 -0}{4-0} (x-0)\\y = \frac{-1}{2} x......(3)\\2y =-x \\x +2y =0$

The first line has an open circle at (0,  -2) and continues up through (-5, 3) with an arrow instead of an endpoint.

The second line has a closed circle at (0, 0) and continues down with a negative slope through (4, -2) with an arrow instead of an endpoint. $f(x) = \dfrac{-1}{2}x \; for\; x < 0$

Hence the piece wise function can be defined by equations (2) and (3)

$f(x) = -x-2 \; for\; x < 0$

$f(x) = \dfrac{-1}{2}x \; for\; x \geq 0$

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