Graph each function. Give the domain and range. g(x)=[[2x-1]]

Verified step by step guidance

1. Identify the function given: \(g(x) = \left\lfloor 2x - 1 \right\rfloor\), where \(\left\lfloor \cdot \right\rfloor\) denotes the floor function, which outputs the greatest integer less than or equal to the input.
2. Determine the domain of \(g(x)\). Since the expression inside the floor function, \$2x - 1\(, is defined for all real numbers, the domain is all real numbers: \)(-\infty, \infty)$.
3. To find the range, consider that the floor function outputs integers. As \(x\) varies over all real numbers, \$2x - 1\( takes all real values from \)-\infty\( to \)\infty\(, so \)g(x)\( takes all integer values from \)-\infty\( to \)\infty\(. Thus, the range is all integers: \)\{ ..., -2, -1, 0, 1, 2, ... \}$.
4. To graph \(g(x)\), note that for each integer \(k\), the function \(g(x) = k\) on the interval where \(k \leq 2x - 1 < k+1\). Solve these inequalities for \(x\) to find the intervals:
5. \[ \frac{k+1}{2} > x \geq \frac{k+1}{2} \]
6. Plot horizontal steps at each integer value \(k\) over the corresponding intervals on the \(x\)-axis, remembering that the graph is a step function with jumps at points where \$2x - 1$ is an integer.

Key Concepts

Greatest Integer Function (Floor Function)

The greatest integer function, denoted by [[x]] or ⌊x⌋, maps any real number to the largest integer less than or equal to that number. For example, [[2.3]] = 2 and [[-1.7]] = -2. Understanding this function is essential for graphing g(x) = [[2x - 1]] because it creates a step-like graph.

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 (y-values). For g(x) = [[2x - 1]], the domain is all real numbers, but the range consists of all integers because the greatest integer function outputs integers only.

Graphing Step Functions

Step functions like the greatest integer function produce graphs with horizontal segments and jumps at integer boundaries. When graphing g(x) = [[2x - 1]], each step corresponds to intervals where 2x - 1 lies between consecutive integers, resulting in a series of horizontal line segments with jumps at specific x-values.