1 Answer

Pavel Shkleinik · Answer 1 · 2020-11-16T16:14:57+0000

Answer:

Shown below

This is a step function which is a type of function defined by two or more functions. To write the function, we can follow these steps:

STEP 1. A linear function.

This function goes from
$-\infty$ to
without including this last value in the domain because at
x=2 we have a hole. By writing the equation:

$The \ equation \ of \ the \ line \ with \ slope \ m \\ passing \ through \ the \ point \ (x_(1),y_(1)) \ is:\\ \\ y-y_(1)=m(x-x_(1))$

So:

$P_(1)(1,0) \\ P_(2)(2,2) \\ \\ y-0=(2-0)/(2-1)(x-1) \\ \\ y=2(x-1) \therefore \boxed{y=2x-2}$

$f(x)=2x-2, \ \ -\infty<x \leq 2$

STEP 2. Constant function.

This function goes from
to
including both values in the domain because at
x=2 and at
x=5 we have dots.

So:

$f(x)=4, \ \ 2 \leq x \leq 5$

STEP 3. Linear function.

This is also a linear function but goes from
to
$+\infty$ without including
x=5 in the domain because we have a hole here. By writing the equation:

So:

$P_(1)(5,6) \\ P_(2)(6,7) \\ \\ y-6=(7-6)/(6-5)(x-5) \\ \\ y=x-5+6 \therefore \boxed{y=x-1}$

$f(x)=x+1, \ \ 5<x \leq +\infty$

Finally, our step function is:

$f(x)=\left\{ \begin{array}{c}2x-2,\;-\infty<x\leq2\\4,\;2\leq x\leq5\\x+1,\;5<x\leq+\infty\end{array}\right.$