on a separate sheet of paper, graph the following function. in the answer box, describe the function…

Explanation:

Step1: Recall the definition of the greatest - integer function

The greatest - integer function $[x]$ gives the greatest integer less than or equal to $x$. For example, $[3.2]=3$, $[-2.7]= - 3$.

Step2: Analyze the transformation of the function

The function $y = f(x)=2[x]-3$ is a transformation of the greatest - integer function $y = [x]$. The factor of 2 vertically stretches the graph of $y = [x]$ by a factor of 2, and the subtraction of 3 shifts the graph of $2[x]$ down by 3 units.

Step3: Consider the behavior on intervals

For any interval $[n,n + 1)$ where $n$ is an integer, $[x]=n$. So, $f(x)=2n - 3$ on the interval $[n,n + 1)$. The graph of $y = f(x)$ consists of a series of horizontal line - segments. At $x=n$, the function value is $2n - 3$, and the line - segment is closed at the left - hand end (because of the $\leq$ in the definition of $[x]$) and open at the right - hand end.

Answer:

The function $f(x)=2[x]-3$ is a step - function. Its graph consists of horizontal line - segments. For each integer $n$, on the interval $[n,n + 1)$, the function has a constant value of $2n - 3$. The graph is a vertically stretched (by a factor of 2) and vertically shifted (down by 3 units) version of the greatest - integer function $y = [x]$.