|x|, known as the absolute value of x, represents the distance between the number x and zero on the number line. It is a fundamental concept in mathematics that describes the magnitude or size of a number, regardless of its sign.
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The absolute value of a number is always non-negative, as it represents the distance from zero.
The absolute value function $f(x) = |x|$ is a piecewise function, defined as $f(x) = x$ for $x \geq 0$ and $f(x) = -x$ for $x < 0$.
The graph of the absolute value function is a V-shaped curve, with the vertex at the origin (0, 0).
The absolute value function is an even function, meaning that $|x| = |-x|$, which results in a symmetric graph about the y-axis.
Absolute value is used in many mathematical contexts, such as distance, inequalities, and defining the domain of functions.
Review Questions
Explain the geometric interpretation of the absolute value function $f(x) = |x|$.
The absolute value function $f(x) = |x|$ can be interpreted geometrically as the distance between the number $x$ and the origin (0, 0) on the number line. The graph of the absolute value function is a V-shaped curve, with the vertex at the origin. This means that for any input $x$, the output $|x|$ represents the absolute distance or magnitude of $x$ from the origin, regardless of the sign of $x$.
Describe the properties of the absolute value function that make it a piecewise function.
The absolute value function $f(x) = |x|$ is a piecewise function because it is defined by different formulas or expressions on different intervals of the domain. Specifically, the function is defined as $f(x) = x$ for $x \geq 0$ and $f(x) = -x$ for $x < 0$. This piecewise definition is necessary to capture the fact that the absolute value of a number is always non-negative, regardless of the sign of the original number.
Analyze the symmetry of the absolute value function and explain its significance in the context of 1.6 Absolute Value Functions.
The absolute value function $f(x) = |x|$ exhibits symmetry about the y-axis, meaning that $|x| = |-x|$. This symmetry is a crucial property in the context of 1.6 Absolute Value Functions, as it allows for the graph of the function to be easily constructed and understood. The symmetric nature of the absolute value function also simplifies the analysis of its behavior, such as the identification of its domain, range, and transformations, which are important topics covered in this chapter.