Question

unit test
unit test complete
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the graph of the step function g(x) = -|x| + 3 is shown. what is the domain of g(x)?
graph of the step function with x-axis from -5 to 5 and y-axis from -5 to 5
options:
○ {x| x is a real number}
○ {x| x is an integer}
○ {x| -2 ≤ x < 5}
○ {x| -1 ≤ x ≤ 5}
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Explanation:

The function \( g(x)=-\lfloor x
floor + 3\) (assuming it's a floor function, a type of step function) or the given step - function graph. The domain of a function is the set of all possible input values (x - values). For a step function like this, looking at the graph, the x - values that the function is defined for are all real numbers? Wait, no, wait. Wait, the graph has points at integer x - values? Wait, no, let's re - examine. Wait, the function is a step function, and the graph is made up of horizontal line segments (steps) at integer x - values? Wait, no, the options: one option is "x is a real number", another is "x is an integer", etc. Wait, the function \( g(x)=-\lfloor x
floor+3\) (floor function) has a domain of all real numbers, because the floor function \(\lfloor x
floor\) is defined for all real numbers x. The floor function takes any real number x and gives the greatest integer less than or equal to x. So the function \( g(x)=-\lfloor x
floor + 3\) is defined for all real numbers x. Let's check the options:

• Option 1: \(\{x|x\text{ is a real number}\}\) – this is correct because the floor function (and thus the function \(g(x)\)) is defined for all real numbers.
• Option 2: \(\{x|x\text{ is an integer}\}\) – the floor function is not only defined for integers, it's defined for all real numbers, so this is wrong.
• Option 3: \(\{x|- 2\leq x\lt5\}\) – the function is defined outside this interval too, for example, x = - 3, x = 6, etc., so this is wrong.
• Option 4: \(\{x|-1\leq x\leq5\}\) – again, the function is defined outside this interval, so this is wrong.

So the domain of \(g(x)\) is all real numbers.

Answer:

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