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Chapter 1: Problem 42

(a) use a graphing utility to graph the function and (b) state the domain and range of the function. $$k(x)=4\left(\frac{1}{2} x-\left[\left[\frac{1}{2} x\right]\right]\right)^{2}$$

Short Answer

Expert verified
The graph of the function \(k(x)=4\left(\frac{1}{2} x-\left[\left[\frac{1}{2}x\right]\right]\right)^{2}\) consists of a series of parabolas with minimum points at each integer x-value due to the presence of the floor function. The domain is \(R\), all real numbers, and the range is [0, 鈭�).

Step by step solution

01

Understand the function

First, recognize the function as a quadratic shifted and truncated by the floor function, which takes the greatest integer less than or equal to a number. This means that it will have repeated 'chunks' for every integer x-value. The domain will be all real numbers, since there aren't any values x can't take in this equation.
02

Graph a single period of the function

For the graph, it would be a parabola opening upwards for each integer value of x, with a minimum at each integer x. This is because the floor function will 'reset' at each integer value, leading to a series of periodic parabolas. The best way to visualize this would be to either graph it by hand for a few periods, or use a graphing calculator or software to plot it. The key is to realize that the function repeats itself every integer x-value.
03

State the domain and range of the function

The domain of the function is \(R\), all real numbers, because there isn't any restriction on what values x can take. The range of the function is [0, 鈭�) because the lowest point on each period of the graph is 0, and the parabola opens upwards.

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