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The absolute value of a number refers to its distance from zero on the number line, regardless of direction. Thus, the absolute value of both positive and negative numbers is always non-negative. This property is crucial when dealing with absolute value functions. In mathematical terms:

• If the number is positive or zero, the absolute value is simply the number itself, thus for any number \( x \), if \( x \geq 0 \), then \(|x| = x\).
• If the number is negative, the absolute value is the opposite of the number, meaning \( |x| = -x \) when \( x < 0 \).

For example, in the function \( f(x) = x + |x| \), the absolute value determines how the function behaves differently for positive and negative values of \( x \). This characteristic leads to different formulas for different domain parts, creating a piecewise function. In our case:

• For \( x \geq 0 \), \( f(x) = x + x = 2x \).
• For \( x < 0 \), \( f(x) = x - x = 0 \).

To graph the function like \( f(x) = x + |x| \), a good starting point is understanding the function behavior split by the absolute value. This function forms a piecewise function, requiring different expressions for different intervals of \( x \).
Firstly, organize the expressions:

• For \( x \geq 0 \), the expression simplifies to \( 2x \), leading to a straight line graph that passes through the origin and has a gradient of 2.
• For \( x < 0 \), it simplifies to 0, resulting in a horizontal line along the x-axis.

This organization helps in plotting the function since each piece is treated separately. Begin by marking key points or plotting lines based on these expressions:

• Draw the horizontal line for \( x < 0 \) clearly along the x-axis.
• Plot the inclined line starting at the origin moving upwards for \( x \geq 0 \) with a slope of 2.

The intersection point where the behavior changes (often where the absolute value function is considered zero) is critical. For this function, the transition is smooth at \( x = 0 \). Understanding these core steps in graphing helps bring clarity to functions involving absolute values and piecewise scenarios.

The slope of a line is a measure of its steepness, often described as the ratio of the vertical change to the horizontal change between two points on the line. It's represented mathematically as \( m \). The slope can tell us how a function is increasing, decreasing, or remaining constant.
For a straight line, the slope \( m \) is calculated as \( m = \frac{\Delta y}{\Delta x} \) where \( \Delta y \) is the change in \( y \) and \( \Delta x \) is the change in \( x \).

• If \( m > 0 \), the line is increasing, rising to the right.
• If \( m < 0 \), the line is decreasing, falling to the right.
• If \( m = 0 \), the line is horizontal, indicating no change in height across the x-axis.

In the given function \( f(x) = x + |x| \), the slope changes:

• For \( x \geq 0 \): The function becomes \( 2x \) with a slope of 2, meaning it rises steeply.
• For \( x < 0 \): The slope is 0 since the function simplifies to \( 0 \), hence it's a flat line on the x-axis.

Understanding the slope provides insight into each section's behavior within a piecewise function. This concept is vital when deciphering how different parts of a function behave, particularly how they incline or stay steady at various intervals.