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

Verified step by step guidance

1. 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.
2. 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.
3. 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.
4. Determine the domain: Since \(x\) can be any real number, the domain is all real numbers, expressed as \((-\infty, \infty)\).
5. 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 \(\{ ..., -2, -1, 0, 1, 2, ... \}\).

Key Concepts

Piecewise and Step Functions

The function ƒ(x) = [[2x]] represents the greatest integer function (floor function), which outputs the greatest integer less than or equal to 2x. This creates a step-like graph where the function value remains constant over intervals and jumps at integer points, making it a piecewise constant function.

Domain and Range of Functions

The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (ƒ(x)-values). For ƒ(x) = [[2x]], the domain is all real numbers, but the range consists of all integers because the floor function outputs integers.

Graphing Floor Functions

To graph a floor function like ƒ(x) = [[2x]], plot horizontal line segments for each integer output value over intervals where 2x lies between consecutive integers. The graph has jump discontinuities at points where 2x is an integer, and the function value steps up by 1 at these points.