Question

what is the domain of the step function $f(x) = \lceil 2x \
ceil - 1$?
$\circ \\ \\{x|\\ x \geq -1\\}$
$\circ \\ \\{x|\\ x \geq 1\\}$
$\circ \\ \\{x|\\ x \text{ is an integer}\\}$
$\circ \\ \\{x|\\ x \text{ is a real number}\\}$

Explanation:

Step1: Recall Domain of Step Functions

Step functions (like ceiling or floor functions) take real numbers as inputs. The ceiling function \(\lceil 2x
ceil\) is defined for any real number \(x\), as multiplying a real number by 2 and taking the ceiling is valid for all real \(x\). Subtracting 1 (a constant) from the ceiling function doesn't restrict the domain further.

Step2: Analyze Options

• Option 1: \(x \geq -1\) is incorrect, as the function is defined for all real numbers, not just \(x \geq -1\).
• Option 2: \(x \geq 1\) is incorrect, same reason as above.
• Option 3: \(x\) being an integer is incorrect, since step functions accept all real numbers (e.g., \(x = 0.5\) gives \(\lceil 1

ceil - 1 = 0\), which is valid).

• Option 4: \(x\) is a real number matches the domain of the step function.

Answer:

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