The modulus function, also known as the absolute value function, is a mathematical function that calculates the distance of a real number from zero on the number line, regardless of the number's sign. It provides a way to measure the magnitude or absolute value of a quantity, without regard to its direction or sign.
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The modulus function is defined as |x|, where x is a real number. It represents the absolute value or distance of x from zero on the number line.
The modulus function always returns a non-negative value, as it measures the distance from zero, regardless of the sign of the input.
Graphically, the modulus function is represented by a V-shaped graph, with the vertex at the origin (0, 0) and the two arms extending infinitely in the positive and negative directions.
The modulus function is an even function, meaning that |x| = |-x|. This is because the distance from zero is the same for a number and its negative counterpart.
The modulus function is commonly used in various mathematical contexts, such as in the definition of distance, in solving equations, and in the analysis of functions.
Review Questions
Explain how the modulus function is used to measure the distance of a real number from zero on the number line.
The modulus function, |x|, calculates the distance of a real number, x, from zero on the number line. Regardless of whether the number is positive or negative, the modulus function will always return a non-negative value that represents the absolute value or magnitude of the number. This allows the modulus function to be used to measure the distance from zero, which is a crucial concept in various mathematical contexts, such as in the definition of distance, solving equations, and analyzing functions.
Describe the graphical representation of the modulus function and explain its properties as an even function.
The graph of the modulus function, |x|, is a V-shaped curve with the vertex at the origin (0, 0). The two arms of the V extend infinitely in the positive and negative directions, reflecting the fact that the modulus function always returns a non-negative value. The modulus function is an even function, meaning that |x| = |-x|. This is because the distance from zero is the same for a number and its negative counterpart, as the modulus function measures the absolute value or magnitude of the input, without regard to its sign.
Analyze how the modulus function is used in various mathematical contexts, such as in the definition of distance, solving equations, and the analysis of functions.
The modulus function is a versatile mathematical tool that is widely used in various contexts. In the definition of distance, the modulus function is used to calculate the absolute value or magnitude of the difference between two points, which is a fundamental concept in geometry and analysis. When solving equations, the modulus function can be used to transform the equation into an equivalent form that is easier to solve. Additionally, the modulus function is often used in the analysis of functions, as it can be used to define and study the properties of functions, such as their symmetry, periodicity, and behavior near critical points.
The absolute value of a number is the distance of that number from zero on the number line, regardless of the number's sign. It is denoted by vertical bars, like |x|.
Number Line: A number line is a visual representation of the set of real numbers, with numbers arranged in a linear fashion from left to right, with zero in the center.