Answer:
Shown below
Step-by-step explanation:
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 [tex]-\infty[/tex] to [tex]2[/tex] without including this last value in the domain because at [tex]x=2[/tex] we have a hole. By writing the equation:
[tex]The \ equation \ of \ the \ line \ with \ slope \ m \\ passing \ through \ the \ point \ (x_{1},y_{1}) \ is:\\ \\ y-y_{1}=m(x-x_{1})[/tex]
So:
[tex]P_{1}(1,0) \\ P_{2}(2,2) \\ \\ y-0=\frac{2-0}{2-1}(x-1) \\ \\ y=2(x-1) \therefore \boxed{y=2x-2}[/tex]
[tex]f(x)=2x-2, \ \ -\infty<x \leq 2[/tex]
STEP 2. Constant function.
This function goes from [tex]2[/tex] to [tex]5[/tex] including both values in the domain because at [tex]x=2[/tex] and at [tex]x=5[/tex] we have dots.
So:
[tex]f(x)=4, \ \ 2 \leq x \leq 5[/tex]
STEP 3. Linear function.
This is also a linear function but goes from [tex]5[/tex] to [tex]+\infty[/tex] without including [tex]x=5[/tex] in the domain because we have a hole here. By writing the equation:
So:
[tex]P_{1}(5,6) \\ P_{2}(6,7) \\ \\ y-6=\frac{7-6}{6-5}(x-5) \\ \\ y=x-5+6 \therefore \boxed{y=x-1}[/tex]
[tex]f(x)=x+1, \ \ 5<x \leq +\infty[/tex]
Finally, our step function is:
[tex]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.[/tex]
