Question

what is the domain of the step function f(x) = ⌈2x⌉ - 1?
○ {x| x ≥ -1}
○ {x| x ≥ 1}
○ {x| x is an integer}
○ {x| x is a real number}

Explanation:

Step1: Recall Domain of Step Functions

The ceiling function (step function here) \(\lceil 2x
ceil\) takes any real number \(x\), multiplies by 2, and gives the least integer greater than or equal to \(2x\). There are no restrictions (like division by zero, square root of negative) on \(x\) here.

Step2: Analyze Each Option

• Option 1: \(x\geq - 1\) is wrong, no lower bound on \(x\) for ceiling function.
• Option 2: \(x\geq1\) is wrong, same reason as above.
• Option 3: \(x\) integer is wrong, ceiling function accepts all reals, not just integers.
• Option 4: \(x\) real number: Since \(2x\) is defined for all real \(x\), and ceiling of any real is defined, the domain is all real numbers.

Answer:

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