Funktionswertbereich

The term "Funktionswertbereich" is a German mathematical expression that translates to "range of a function" in English. It refers to the set of all possible output values that a function can produce when its input values (domain) are varied over all permissible values.

In mathematical analysis, the range is an essential aspect of understanding a function's behavior and properties. It helps to identify the extent of the output values and how the function maps inputs to outputs. The range is often determined through various methods, such as algebraic manipulation, graphical analysis, or calculus techniques like finding critical points and examining the behavior at the boundaries.

For example, consider the function f(x) = x², where the domain is all real numbers. The function's range is the set of all non-negative real numbers, expressed as [0, ∞). This indicates that no matter which real number is input into the function, the output will always be zero or positive.

In the context of functions with restricted domains, the range may be smaller or different from the overall possible output values of a similar, unbounded function. Determining the Funktionswertbereich is crucial in various applications, including solving equations, optimization problems, and modeling real-world phenomena.

The concept of the Funktionswertbereich also plays a significant role in advanced mathematics, including calculus, where techniques such as the use of derivatives are employed to find the maximum and minimum values of a function, thus identifying its maximum and minimum in the range.

Overall, the Funktionswertbereich is a fundamental element in understanding and analyzing the output behavior of functions in mathematics.