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

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Answer 1 · 2018-03-01T08:29:06

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

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

# "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." #