# What is the domain of the step function f(x)=[2x-1]?

Mar 1, 2018

$\setminus \quad \text{The answer, in interval notation, and in set notation, is:}$

$\setminus q \quad \setminus q \quad \setminus \quad \setminus q \quad \setminus \quad \setminus \text{domain of} \setminus \setminus f \left(x\right) \setminus = \setminus \left(- \infty , \infty\right) \setminus = \setminus \mathbb{R} .$

#### Explanation:

$\text{We want the domain of the step function:} \setminus q \quad f \left(x\right) \setminus = \setminus \left[2 x - 1\right] .$

$\text{1) The linear function," \ 2 x - 1, "is defined for all real numbers,}$
$\setminus q \quad \setminus \quad x .$

$\text{2) The greatest integer function," \ [ x ], \ "is defined for all real}$
$\setminus q \quad \setminus \quad \text{numbers,} \setminus \setminus x .$

$\text{3) So now, in language that should be clear:}$

 \qquad \qquad \qquad \qquad f( "real" ) \ = \ [ \ 2("real") -1 \ ] \ = \ [ \ "real" \ ] \ = \ "real".

$\text{4) Thus:}$

 \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad f( "real" ) \ = \ "real".

$\setminus q \quad \setminus \quad \text{So:}$

$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus f \left(x\right) \setminus \quad \text{is defined for all real numbers} \setminus \setminus x .$

$\setminus q \quad \setminus \quad \text{Thus, in interval notation, and in set notation:}$

$\setminus q \quad \setminus q \quad \setminus \quad \setminus q \quad \text{domain of} \setminus \quad f \left(x\right) \setminus = \setminus \left(- \infty , \infty\right) \setminus = \setminus \mathbb{R} .$

$\text{This is our answer.}$