Domain, Graph, And Range Of F(x) = √(x-3)

In the realm of mathematics, understanding the domain, graph, and range of a function is crucial for comprehending its behavior and characteristics. In this article, we will delve into the function f(x) = √(x-3), a square root function, and explore its domain, which represents the set of all possible input values (x-values) for which the function is defined; its graph, which visually depicts the function's behavior on the coordinate plane; and its range, which encompasses all possible output values (y-values) that the function can produce. By examining these aspects, we will gain a comprehensive understanding of this function's properties and its graphical representation. Understanding the function's domain, range, and graph is foundational for various mathematical applications, including solving equations, modeling real-world phenomena, and analyzing data. This exploration will provide a solid foundation for further studies in mathematics and related fields.

The domain of a function is the set of all possible input values (x-values) for which the function produces a real number output. For the function f(x) = √(x-3), we need to consider the restriction imposed by the square root. The square root of a negative number is not a real number, so we must ensure that the expression inside the square root, (x-3), is greater than or equal to zero. In essence, the domain represents the permissible x-values that won't result in an undefined output, and in this case, we're dealing with the square root function, where the radicand (the expression inside the square root) must be non-negative. When determining the domain, it is essential to identify any restrictions on the input values that would lead to undefined or non-real outputs. This includes avoiding negative values under square roots, division by zero, and logarithms of non-positive numbers. Therefore, to find the domain, we set up the inequality:

x - 3 ≥ 0

Solving this inequality for x, we get:

x ≥ 3

This inequality tells us that the function is defined for all x-values greater than or equal to 3. In interval notation, we represent this domain as [3, ∞). This notation indicates that the domain includes all real numbers from 3 (inclusive) to positive infinity. The left square bracket '[' signifies that 3 is included in the domain, while the parenthesis ')' indicates that infinity is not a specific number and is not included. The domain is a fundamental concept in mathematics as it defines the scope within which the function operates, and understanding it is crucial for interpreting the function's behavior and making valid calculations.

To graph the function f(x) = √(x-3), we can start by creating a table of values. We'll choose x-values within the domain we found earlier (x ≥ 3) and calculate the corresponding y-values. Several key points are essential for accurately graphing a square root function. First, we need to identify the starting point, which is the point where the square root function begins. In this case, the starting point is (3, 0), as this is the smallest x-value in the domain. Next, we can plot additional points by choosing x-values greater than 3 and calculating the corresponding y-values. These points will help us sketch the curve of the function. We can also consider the general shape of the square root function, which is a curve that starts at a point and extends in one direction. By combining these techniques, we can create an accurate and informative graph of the function f(x) = √(x-3). Here are a few points:

• When x = 3, f(3) = √(3-3) = √0 = 0
• When x = 4, f(4) = √(4-3) = √1 = 1
• When x = 7, f(7) = √(7-3) = √4 = 2
• When x = 12, f(12) = √(12-3) = √9 = 3

Plotting these points (3, 0), (4, 1), (7, 2), and (12, 3) on a coordinate plane, we can see that the graph starts at the point (3, 0) and curves upwards and to the right. The graph is a transformation of the basic square root function y = √x, shifted 3 units to the right. The shifting of the graph is a crucial aspect of understanding function transformations, and recognizing this shift allows us to quickly sketch the graph without needing to plot numerous points. This rightward shift corresponds to the subtraction of 3 within the square root, illustrating a fundamental principle of horizontal transformations in functions. The graph provides a visual representation of the function's behavior, allowing us to quickly grasp its key characteristics, such as its increasing nature and its starting point.

The range of a function is the set of all possible output values (y-values) that the function can produce. For the function f(x) = √(x-3), we can determine the range by analyzing the graph or by considering the properties of the square root function. Looking at the graph, we see that the function starts at y = 0 and extends upwards without bound. This indicates that the range includes all non-negative real numbers. A thorough comprehension of the range is vital for understanding the function's output behavior and its applicability in modeling real-world situations. For a square root function, the range is inherently restricted to non-negative values because the square root of a real number cannot be negative. This intrinsic property of square root functions informs our analysis and allows us to deduce the range with confidence. We know that the square root of any non-negative number is also non-negative. Since x-3 is always non-negative within the domain [3, ∞), the square root √(x-3) will also be non-negative. The smallest possible value of √(x-3) is 0, which occurs when x = 3. As x increases, the value of √(x-3) also increases without bound. Therefore, the range of the function is all non-negative real numbers. In interval notation, we represent the range as [0, ∞). This notation indicates that the range includes all real numbers from 0 (inclusive) to positive infinity. The left square bracket '[' signifies that 0 is included in the range, while the parenthesis ')' indicates that infinity is not a specific number and is not included. The range, together with the domain, provides a comprehensive picture of the function's behavior, allowing us to predict and interpret its output for any valid input.

In conclusion, we have explored the function f(x) = √(x-3), determining its domain to be [3, ∞), sketching its graph which is a square root function shifted 3 units to the right, and identifying its range as [0, ∞). By understanding the domain, graph, and range, we gain a complete understanding of the function's behavior and characteristics. This knowledge is essential for various mathematical applications and provides a strong foundation for further studies in mathematics. Understanding the domain, graph, and range is a fundamental skill in mathematics, enabling us to analyze and interpret functions effectively. These concepts are essential for solving equations, modeling real-world phenomena, and making predictions based on mathematical models. By mastering these skills, we empower ourselves to tackle more complex mathematical challenges and apply mathematical principles to a wide range of disciplines.