Question

Graph each function. Give the domain and range. ƒ(x)=[[2x]]

Accepted Answer

Identify the function given: $$f(x) = \left\lfloor 2x \right\rfloor$$, where $$\left\lfloor \cdot \right\rfloor$$ denotes the floor function, which outputs the greatest integer less than or equal to the input. Understand the behavior of the floor function: For any real number input, $$\left\lfloor 2x \right\rfloor$$ will 'step' down to the nearest integer. This creates a step graph with jumps at points where $$2x is an $$integer. To graph the function, choose several values of $$x$$ and calculate $$f(x) by $$multiplying $$x by 2 $$and then applying the floor function. For example, for $$x=0$$, $$f(0) = \left\lfloor 0 \right\rfloor = 0$$; for $$x=0.5$$, $$f(0.5) = \left\lfloor 1 \right\rfloor = 1$$; and so on. Determine the domain: Since $$x$$ can be any real number, the domain is all real numbers, expressed as $$(-\infty, \infty). $$Determine the range: Because $$f(x)$$ takes all integer values (as $$2x$$ covers all real numbers and the floor function outputs integers), the range is all integers, expressed as $$\{ ..., -2, -1, 0, 1, 2, ... \}.$$

Graph each function. Give the domain and range. ƒ(x)=[[2x]]

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Key Concepts

Piecewise and Step Functions

Domain and Range of Functions

Graphing Floor Functions