Question

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

Answer 1

`\quad "The answer, in interval notation, and in set notation, is:"`

`\qquad \qquad \quad \qquad \quad \ "domain of" \ \ f(x) \ = \ ( - infty, infty ) \ = \ RR.`

Explanation:

`"We want the domain of the step function:" \qquad f(x) \ = \ [ 2 x - 1 ].`

`"1) The linear function," \ 2 x - 1, "is defined for all real numbers,"`
`\qquad \quad x.`

`"2) The greatest integer function," \ [ x ], \ "is defined for all real"`
`\qquad \quad "numbers," \ \ x.`

`"3) So now, in language that should be clear:"`

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

`"4) Thus:"`

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

`\qquad \quad "So:"`

`\qquad \qquad \qquad \qquad \qquad \ f(x) \quad "is defined for all real numbers" \ \ x.`

`\qquad \quad "Thus, in interval notation, and in set notation:"`

`\qquad \qquad \quad \qquad "domain of" \quad f(x) \ = \ ( - infty, infty ) \ = \ RR.`

`"This is our answer."`