Problem 32 - The Calculus with Analytic Geometry

Question

In Exercises 11 through 34, the function is the set of all ordered pairs \((x, y)\) satisfying the given equation. Find the domain and range of the function, and draw a sketch of the graph of the function. $$ f: y=(x-\llbracket x \rrbracket)^{2} $$

Step-by-Step Solution

Answer

Domain: all real numbers. Range: [0, 1). Sketch: Series of upward parabolas within each integer interval (n, n+1).

Step 1: Understand the Given Function

The function provided is \( f: y=(x-\llbracket x \rrbracket)^{2} \). Here, \( \llbracket x \rrbracket \)) represents the greatest integer function, also known as the floor function, which returns the greatest integer less than or equal to \(x\).

Step 2: Determine the Domain

The domain of a function is the set of all possible input values (\(x\)). Since \(x\) can be any real number and the floor function \( \llbracket x \rrbracket \) is defined for all real numbers, the domain of this function is all real numbers (\( -\infty < x < \infty \)).

Step 3: Analyze the Output (Range)

To find the range, observe that \(y=(x-\llbracket x \rrbracket)^{2}\) represents the square of the fractional part of \(x\). The fractional part of any number \(x\) lies in the interval \([0, 1)\), so \( 0 \leq x-\llbracket x \rrbracket < 1.\) Squaring this, we get \(0 \leq (x-\llbracket x \rrbracket)^{2} < 1.\) Therefore, the range of the function is \([0, 1)\).

Step 4: Sketch the Graph of the Function

To sketch the graph of \( y = (x - \llbracket x \rrbracket)^2 \), note that for each integer interval \( [n, n+1) \) where \( n \) is an integer, the graph will look like a parabola opening upwards with its vertex at \((n, 0)\). The function repeats this pattern for all integer values of \( n \). Draw these parabolic segments for multiple intervals to show the periodic nature of the function.

Key Concepts

Domain and Range

The domain of a function is all the possible input values (usually noted as x) that the function can accept. For our function, \(f: y = (x-\llbracket x \rrbracket)^{2}\), the domain is straightforward because the greatest integer function \( \llbracket x \rrbracket\) is defined for all real numbers. Thus, the domain of our function is all real numbers: \(-\infty < x < \infty \).

The range of a function is the set of all possible output values. In this case, our output, or y, depends on \( (x - \llbracket x \rrbracket)^{2}\). The term \(x - \llbracket x \rrbracket \) represents the fractional part of \(x\), which lies between 0 and 1, but does not include 1. When we square any number in this interval, the result will always be between 0 and 1, but not including 1. Therefore, the range of this function is \( [0, 1) \).

If we summarize:

• Domain: All real numbers \( -\infty < x < \infty \)
• Range: [0, 1)

Graphing Functions

When graphing a function, you'll want to visualize how the output \(y\) changes with different inputs \(x\). For our function, \( y = (x - \llbracket x \rrbracket)^2 \), the key step is understanding the pattern within each interval \( [n, n+1) \) where \( n \) is an integer.

Because \( x - \llbracket x \rrbracket \) gives us the fractional part of \( x \), within each interval, the graph resembles a parabola that opens upwards. The vertex of each parabolic segment is at the integer \( n \), starting from \( y = 0 \) and reaching close to but not touching \( y = 1 \). This pattern repeats for every integer interval.

To draw the graph:

• For each interval \( [n, n+1) \), start your parabola at \( (n, 0) \) and plot up to just before \( (n+1, 1) \).
• Move to the next integer and repeat the process.

This way, you’ll see a continuous and periodic pattern of parabolas, each within their respective integer intervals.

Greatest Integer Function

The greatest integer function, often represented by \( \llbracket x \rrbracket \), returns the greatest integer less than or equal to \( x \). It’s commonly known as the floor function. For example:

• \( \llbracket 3.7 \rrbracket = 3 \)
• \( \llbracket -2.5 \rrbracket = -3 \)
• \( \llbracket 4 \rrbracket = 4 \)

This function is particularly useful in calculus and graphing because it helps to break down continuous functions into manageable integer-based segments.

By understanding and applying the greatest integer function, it becomes easier to graph functions that involve integer bounds and to find the patterns in functions with periodic behavior, like in our function \( y = (x-\llbracket x \rrbracket)^{2} \). The floor function helps in identifying the locations where the function resets its behavior, exemplified by the intervals \( [n, n+1) \).