Question

Which function is graphed below?

Answer 1

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 $-\infty$ to $2$ 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:

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

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

STEP 2. Constant function.

This function goes from $2$ to $5$ 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 $5$ to $+\infty$ without including $x=5$ in the domain because we have a hole here. By writing the equation:

So:

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

$f(x)=x+1, \ \ 5$

Finally, our step function is:

$f(x)=\left\{ \begin{array}{c}2x-2,\;-\infty$