Domain And Range Of F(x) = √(x-1) In Interval Notation

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In the realm of mathematics, functions serve as fundamental building blocks for modeling and understanding relationships between variables. Among the diverse family of functions, the square root function holds a special place, often encountered in various mathematical contexts and real-world applications. This article delves into the intricacies of the function f(x) = √(x-1), with a primary focus on determining its domain and range using interval notation. Understanding the domain and range of a function is crucial as it unveils the set of permissible input values (domain) and the corresponding set of output values (range). These concepts are pivotal for comprehending the behavior and characteristics of the function.

Delving into the Domain of f(x) = √(x-1)

The domain of a function encompasses all the possible input values (x-values) for which the function produces a valid output. In the context of the square root function, a critical constraint arises from the fact that the square root of a negative number is not defined within the realm of real numbers. Consequently, to ensure a real-valued output for f(x) = √(x-1), the expression under the square root, (x-1), must be greater than or equal to zero.

To determine the domain, we set up the inequality:

x - 1 ≥ 0

Solving for x, we add 1 to both sides:

x ≥ 1

This inequality reveals that the domain of f(x) = √(x-1) consists of all real numbers greater than or equal to 1. Expressing this in interval notation, we use a square bracket '[' to indicate that 1 is included in the domain and the infinity symbol '∞' to represent the unbounded upper limit. Therefore, the domain of f(x) = √(x-1) is:

Domain: [1, ∞)

This interval notation signifies that the function accepts any input value starting from 1 and extending infinitely in the positive direction. In essence, the domain represents the set of x-values for which the function is defined and produces a real-valued output.

Exploring the Range of f(x) = √(x-1)

The range of a function comprises the set of all possible output values (y-values) that the function can generate. To determine the range of f(x) = √(x-1), we consider the behavior of the square root function and the effect of the transformation applied to the input.

The square root function, √x, by its very definition, always produces non-negative values. This fundamental property forms the basis for determining the range of f(x) = √(x-1). The transformation (x-1) inside the square root shifts the graph of the function one unit to the right, but it does not alter the non-negative nature of the output.

As x varies within the domain [1, ∞), the expression (x-1) takes on values from 0 to infinity. Consequently, the square root of (x-1) also varies from 0 to infinity. This implies that the range of f(x) = √(x-1) consists of all non-negative real numbers.

In interval notation, we represent the range using a square bracket '[' to include 0 and the infinity symbol '∞' to denote the unbounded upper limit. Therefore, the range of f(x) = √(x-1) is:

Range: [0, ∞)

This interval notation signifies that the function can produce any output value starting from 0 and extending infinitely in the positive direction. The range essentially captures the spectrum of y-values that the function can attain.

Visualizing the Domain and Range

To further solidify our understanding of the domain and range, let's visualize the graph of f(x) = √(x-1). The graph originates at the point (1, 0) and extends infinitely in the positive x and y directions. This visual representation clearly illustrates that the function is defined for x-values greater than or equal to 1 (domain) and produces y-values greater than or equal to 0 (range).

The graph serves as a powerful tool for confirming our analytical determination of the domain and range. It provides a visual confirmation of the permissible input values and the corresponding output values that the function can generate.

Connecting Domain and Range to Function Behavior

The domain and range of a function provide valuable insights into its behavior and characteristics. In the case of f(x) = √(x-1), the domain [1, ∞) indicates that the function is only defined for x-values greater than or equal to 1. This implies that the function does not exist for x-values less than 1.

The range [0, ∞) reveals that the function's output values are always non-negative. This is a direct consequence of the square root function's inherent property of producing non-negative results. The range also indicates that the function can attain any non-negative value, suggesting that it increases without bound as x increases.

Understanding the domain and range allows us to make informed predictions about the function's behavior and its applicability in various mathematical and real-world contexts.

Practical Applications of Domain and Range

The concepts of domain and range extend far beyond theoretical mathematics, finding practical applications in diverse fields. Consider a scenario where we are modeling the distance traveled by an object as a function of time using f(x) = √(x-1), where x represents time in seconds and f(x) represents the distance in meters.

In this context, the domain [1, ∞) implies that the object starts moving at time t = 1 second, as time cannot be negative. The range [0, ∞) indicates that the distance traveled by the object is always non-negative and can increase indefinitely as time progresses.

This example highlights how the domain and range provide meaningful interpretations within the context of a real-world problem. They help us understand the limitations and possibilities of the mathematical model, ensuring that our predictions align with physical reality.

Conclusion

In this comprehensive exploration, we have unveiled the domain and range of the function f(x) = √(x-1) using interval notation. We determined that the domain is [1, ∞), encompassing all real numbers greater than or equal to 1, and the range is [0, ∞), comprising all non-negative real numbers.

Understanding the domain and range of a function is paramount for comprehending its behavior, characteristics, and applicability. These concepts provide essential insights into the permissible input values and the corresponding output values that the function can generate.

By mastering the techniques for determining domain and range, you equip yourself with a powerful tool for analyzing and interpreting functions in various mathematical and real-world scenarios. This knowledge serves as a cornerstone for further exploration into the fascinating world of functions and their applications.

  • Domain and Range
  • Interval Notation
  • Square Root Function
  • Function Analysis
  • Mathematical Functions
  • f(x) = √(x-1)