Question

what is the domain of the step function $f(x) = \lfloor 2x \
floor - 1$?
$\bigcirc \\ \\{x\mid x \geq -1\\}$
$\bigcirc \\ \\{x\mid x \geq 1\\}$
$\bigcirc \\ \\{x\mid x \text{ is an integer}\\}$
$\bigcirc \\ \\{x\mid x \text{ is a real number}\\}$

Explanation:

Step1: Recall Domain of Step Function

A step function (greatest integer function, \(\lfloor x
floor\)) is defined for all real numbers. For \(f(x)=\lfloor 2x
floor - 1\), the inner function \(2x\) is defined for all real \(x\), and the greatest integer function \(\lfloor 2x
floor\) takes any real number \(2x\) (hence any real \(x\)) and outputs an integer. So there are no restrictions on \(x\) being a real number.

Step2: Analyze Options

• Option 1: \(x\geq - 1\) is wrong, as \(x\) can be less than -1 (e.g., \(x=-2\), \(f(-2)=\lfloor - 4

floor - 1=-4 - 1=-5\), which is defined).

• Option 2: \(x\geq1\) is wrong, \(x = 0\) is valid (\(f(0)=\lfloor0

floor - 1=-1\)).

• Option 3: \(x\) integer is wrong, \(x = 0.5\) is valid (\(f(0.5)=\lfloor1

floor - 1=0\)).

• Option 4: \(x\) real number is correct, as step function is defined for all real numbers.

Answer:

\(\{x|x\text{ is a real number}\}\)