Modulus functions measure the distance of numbers from zero on the number line, always yielding non-negative results. They are represented as f(x) = |x| and have a 'V' shaped graph. Understanding their properties is key to solving related equations and inequalities. In calculus, the derivative and integration of modulus functions require careful consideration due to their piecewise nature. The inverse of a modulus function also demands domain restriction.
Show More
Modulus functions, represented as f(x) = |x|, measure the distance of a number from zero on the number line
Modulus functions are always non-negative, as they measure distance from zero
Modulus functions are distinct from absolute value functions, although they share similar properties
The graph of a modulus function is constructed by reflecting segments below the x-axis upwards
The resulting graph of a modulus function typically has a 'V' shape, with the vertex representing the point where the function changes from negative to positive
The reflection process in graphing modulus functions visualizes the absolute value operation, showing the distance from zero
The modulus of any real number is non-negative
The modulus of a number and its negative is the same
For multiplication and division, the modulus of the product or quotient is the product or quotient of the moduli, respectively
Equations involving modulus functions require considering both the positive and negative cases of the variable expression
Solving inequalities with modulus functions involves graphing and comparing the function to the other expression in the inequality
Modulus functions are piecewise-defined, requiring separate operations for positive and negative intervals in calculus and integration
00
While the ______ of a modulus function includes all real numbers, its ______ consists only of non-negative real numbers.
domain
range
01
Modulus function graph shape
V-shaped due to absolute value operation
02
Vertex of modulus function graph
Point where function switches from negative to positive
