Problem 22 - Calculus Single Variable

Question

Plot several points, and sketch the graph of the function defined by the given expression. $$ \operatorname{signum}\left(\left|x^{2}-x\right|\right) $$

Step-by-Step Solution

Answer

The function is 1 for \( x \neq 0 \) and \( x \neq 1 \), and is 0 at \( x = 0 \) and \( x = 1 \).

Step 1: Understand the Signum Function

The signum function, or \( \operatorname{signum}(x) \), is a piecewise function that gives the sign of a real number. Specifically, \( \operatorname{signum}(x) = -1 \) if \( x < 0 \), \( 0 \) if \( x = 0 \), and \( 1 \) if \( x > 0 \). We will apply this to the expression \( |x^2 - x| \).

Step 2: Examine \( |x^2 - x| \)

First, solve for when \( x^2 - x = 0 \) to understand where the absolute value expression is zero. Factoring gives \( x(x-1) = 0 \), so \( x = 0 \) or \( x = 1 \). Therefore, \( |x^2 - x| = 0 \) at these points, turning the signum function to zero at \( x = 0 \) and \( x = 1 \).

Step 3: Determine Sign on Intervals

To find where \( x^2 - x \) is positive or negative, examine the sign of \( x(x-1) \):- For \( x < 0 \), both factors are negative, so \( x^2 - x > 0 \).- For \( 0 < x < 1 \), \( x > 0 \) and \( x-1 < 0 \), resulting in \( x^2 - x < 0 \).- For \( x > 1 \), both factors are positive, so \( x^2 - x > 0 \).

Step 4: Apply Signum Function

For \( x < 0 \), \( x^2 - x > 0 \) so \( \operatorname{signum}(|x^2 - x|) = 1 \).For \( 0 < x < 1 \), \( x^2 - x < 0 \) but since the absolute value is non-negative, \( |x^2 - x| > 0 \) making \( \operatorname{signum}(|x^2 - x|) = 1 \).For \( x = 0 \) or \( x = 1 \), \( |x^2 - x| = 0 \), hence \( \operatorname{signum}(|x^2 - x|) = 0 \).For \( x > 1 \), \( x^2 - x > 0 \) so \( \operatorname{signum}(|x^2 - x|) = 1 \).

Step 5: Sketch the Graph

From the previous steps, construct the graph:- For \( x < 0 \) and \( x > 1 \), graph the line \( y = 1 \).- For \( x = 0 \) and \( x = 1 \), graph the point \( y = 0 \) as a dot.- For \( 0 < x < 1 \), graph the line \( y = 1 \). This results in a horizontal line at \( y = 1 \) everywhere except at \( x = 0 \) and \( x = 1 \).

Key Concepts

Piecewise Functions

The signum function is a classic example of a piecewise function. Piecewise functions are defined by different expressions or rules for different intervals of the domain. The signum function, denoted as \( \operatorname{signum}(x) \), specifies the sign of a real number. It can take three different values depending on the condition of \( x \):

• If \( x < 0 \), the function returns \(-1\).
• If \( x = 0 \), it returns \(0\).
• If \( x > 0 \), the function returns \(1\).

Understanding these intervals is crucial when plotting piecewise functions because each interval may have a different graph appearance. In the expression given in the problem \( \operatorname{signum}(|x^2 - x|) \), once \( |x^2 - x| \) is calculated, the signum function is applied based on the resulting value.

Absolute Value

The absolute value is a fundamental mathematical concept that measures the distance of a number from zero on the number line, regardless of direction. It is always non-negative. For any real number \( x \), the absolute value is represented by \( |x| \) and is defined as follows:

• \( |x| = x \) if \( x \ge 0 \)
• \( |x| = -x \) if \( x < 0 \)

In the exercise expression \( |x^2 - x| \), the absolute value ensures that \( x^2 - x \) is non-negative. Factoring \( x^2 - x \) gives \( x(x-1) \), which has roots at \( x = 0 \) and \( x = 1 \). Thus, \( |x^2 - x| \) equals zero when \( x \) is either zero or one. In any interval between these roots, the sign of \( x^2 - x \) varies based on \( x \), but the absolute value renders it non-negative, critical for evaluating the subsequent signum function.

Graphing Functions

Graphing functions involves plotting points or lines on a coordinate system to represent the function visually. Understanding how to graph piecewise functions like the signum function is essential:

• Identify the intervals of the function where different conditions apply.
• Determine the behavior of the function in each interval.
• Plot points or lines for each segment based on these conditions.

In this particular exercise, the significant steps include analyzing \( \operatorname{signum}(|x^2 - x|) \) at different intervals:- From \( x < 0 \) and \( x > 1 \), graph \( y = 1 \).- At \( x = 0 \) and \( x = 1 \), plot the point \( y = 0 \).- From \( 0 < x < 1 \), graph \( y = 1 \).This approach results in a piecewise horizontal line at specific levels on the graph.

Quadratic Equations

Quadratic equations are polynomial equations of the form \( ax^2 + bx + c = 0 \). In the exercise, the expression \( x^2 - x \) is a simple quadratic equation, which can be solved to find the roots, an integral part of understanding the function's behavior.

• Factor the quadratic as \( x(x-1) = 0 \).
• The solutions are \( x = 0 \) and \( x = 1 \), which are the points where the quadratic expression hits zero.

These roots divide the number line into intervals: \( x < 0 \), \( 0 < x < 1 \), and \( x > 1 \). Testing the sign of \( x^2 - x \) in these intervals helps determine the value for the absolute value expression, which subsequently dictates the value of the signum function. Quadratic functions like \( x^2 - x \) are fundamental in many areas of mathematics and play a key role in understanding more complex expressions and equations.