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Chapter 1: Problem 34

Sketch the graph of the function. $$g(x)=[[x-3]]$$

Short Answer

Expert verified
To sketch the step function \(g(x) = [[x-3]]\), plot a few points (1,-2), (2,-1), (3,0), (4,1), (5,2) and draw horizontal lines within each interval, leaving an open circle at the end of each right interval.

Step by step solution

01

Understand the Function

The function \(g(x) = [[x-3]]\) is a step function. The inner bracket \([x-3]\) indicates that subtraction 3 from the input x while the outer bracket \([\)] changes the value to the integer part of it. The function will be equal to a integer value n in the interval where n 鈮� x < n + 1.
02

Identify the Steps

Based on the definitions, this function will have next value at every integer x. To find the intervals and corresponding values, we can iterate over a few integers. Suppose let's take from x = 1 to 5, then we get \([1-3] = [-2] = -2, [2-3] = [-1] = -1, [3-3] = [0] = 0, [4-3] = [1] = 1, [5-3] = [2] = 2\). Now we have five points with (x,g(x)) as follows: (1,-2), (2,-1), (3,0), (4,1), (5,2).
03

Sketch the Graph

Plot this on xy-plane. For every interval between two neighboring points, draw a horizontal line at the level of g(x). At the exact integer points, the function jumps to the next value. Therefore, those points should be open circles. Remember, the graph approaches the integer points from the left side, so the lines are solid on the left but not on the right.

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