Question

a step function h(x) is represented by y = -2⌊x⌋. which phrase best describes the range of the function h(x)?
○ all real numbers
○ all rational numbers
○ all negative integers
○ all even integers

Explanation:

Step1: Recall the floor function

The floor function $\lfloor x
floor$ gives the greatest integer less than or equal to $x$. So $\lfloor x
floor$ is an integer (..., -2, -1, 0, 1, 2, ...).

Step2: Analyze the function $y = -2\lfloor x

floor$
Let $n=\lfloor x
floor$, where $n$ is an integer. Then $y = - 2n$. When $n$ is an integer, $-2n$ will be an even integer (since multiplying an integer by 2 gives an even number, and the negative sign just changes the sign, still keeping it an even integer). For example, if $n = 1$, $y=-2$; if $n=-1$, $y = 2$; if $n = 0$, $y = 0$; if $n = 2$, $y=-4$; if $n=-2$, $y = 4$, etc.

• It's not all real numbers because $y$ takes only integer values (specifically even integers).
• It's not all rational numbers because $y$ is restricted to even integers, not all rational numbers.
• It's not all negative integers because when $n$ is negative, $y$ is positive (e.g., $n=-1$, $y = 2$), so it includes positive even integers too.

Answer:

D. all even integers