Question

Graph each function. Give the domain and range. ƒ(x)=-[[x]]

Accepted Answer

Identify the function given: $$f(x) = -\lfloor x \rfloor$$, where $$\lfloor x \rfloor$$ denotes the greatest integer less than or equal to $$x ($$the floor function). Understand the behavior of the floor function: for any real number $$x$$, $$\lfloor x \rfloor is an $$integer that steps up by 1 at each integer value of $$x. $$Since the function is $$-\lfloor x \rfloor$$, the graph will reflect the floor function values across the x-axis (negated). To graph the function, plot points for several values of $$x$$, especially around integers. For example, for $$x in $$the interval $$[0,1)$$, $$\lfloor x \rfloor = 0$$, so $$f(x) = 0. $$For $$x in$$ $$[1,2)$$, $$\lfloor x \rfloor = 1$$, so $$f(x) = -1$$, and so on. Connect these points as step segments that jump at integer values of $$x. $$Determine the domain: since the floor function is defined for all real numbers, the domain of $$f(x) is $$all real numbers, expressed as $$(-\infty, \infty). $$Determine the range: because $$\lfloor x \rfloor$$ takes all integer values from $$-\infty to$$ $$\infty$$, and $$f(x) = -\lfloor x \rfloor$$, the range is also all integers but negated, which is still all integers $$(-\infty, \infty).$$

Graph each function. Give the domain and range. ƒ(x)=-[[x]]

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Key Concepts

Greatest Integer Function (Floor Function)

Function Transformation (Reflection)

Domain and Range of Step Functions