ISI2015-DCG-50

Question

The piecewise linear function for the following graph is

 

• A. $f(x) = \begin{cases} = x, \: x \leq -2 \\ =4, \: -2<x<3 \\ =x+1, \: x \geq 3 \end{cases}$
• B. $f(x) = \begin{cases} = x-2, \: x \leq -2 \\ =4, \: -2<x<3 \\ =x-1, \: x \geq 3 \end{cases}$
• C. $f(x) = \begin{cases} = 2x, \: x \leq -2 \\ =x, \: -2<x<3 \\ =x+1, \: x \geq 3 \end{cases}$
• D. $f(x) = \begin{cases} = 2-x, \: x \leq -2 \\ =4, \: -2<x<3 \\ =x+1, \: x \geq 3 \end{cases}$

Answer 1

For the interval $\left ( -2,3 \right )$,  it is pretty clear that  $f\left ( x \right )=4$.  So option C is eliminated.

Using the two-point form of straight line concept, i.e. equation of a straight line passing through two given points is :  

$y-y_{1}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\left ( x-x_{1} \right )$

So, for the interval  $\left [ -\infty ,2 \right ]$,  the equation of the line would be  $y-4=\left ( -1 \right )\left ( x+2 \right )$  $\Rightarrow$  $y=2-x$

$\therefore$  Option D is the correct answer.