Find The Domain Of Y=4√(x-8) A Step-by-Step Guide

In mathematics, particularly when dealing with functions, the domain is a fundamental concept. The domain of a function represents the set of all possible input values (often denoted as 'x') for which the function produces a valid output. In simpler terms, it's the range of x-values that you can plug into the function without causing any mathematical errors or undefined results. Determining the domain of a function is a crucial step in understanding its behavior and characteristics. For instance, consider the function y = 4√(x-8), which involves a square root. We need to identify the values of x that will result in a real number output. This is because the square root of a negative number is not defined within the realm of real numbers. Similarly, functions may have restrictions due to division by zero or logarithms of non-positive numbers. Therefore, finding the domain involves identifying these restrictions and expressing the valid input values. The domain can be expressed in several ways, including interval notation, set notation, and graphically on a number line. Interval notation uses parentheses and brackets to indicate whether the endpoints are included or excluded. Set notation uses curly braces to list the elements or to define the set using a condition. Graphically, the domain can be represented as a shaded region on the x-axis. Understanding the domain is essential for graphing functions, solving equations, and applying functions in real-world contexts. It helps us to avoid errors and to interpret the results meaningfully. Moreover, the domain often provides insights into the practical limitations of a model represented by the function. For example, if a function models the population growth of a species, the domain might be restricted to non-negative values since negative populations are not physically possible. In summary, the domain is a foundational concept in mathematics that determines the scope of input values for which a function is valid, playing a critical role in both theoretical analysis and practical applications. Understanding the domain helps us to work with functions accurately and to interpret their results effectively.

The function we're tasked with analyzing is y = 4√(x-8). This function is a transformation of the basic square root function, y = √x. The presence of the square root introduces a critical restriction on the domain, as we cannot take the square root of a negative number within the set of real numbers. This means that the expression inside the square root, (x-8) in this case, must be greater than or equal to zero. The coefficient 4 in front of the square root does not affect the domain but rather scales the output of the function. To fully understand this function, it's helpful to break down its components and consider how they interact. The square root function itself, √x, is a fundamental mathematical function that maps non-negative real numbers to their non-negative square roots. Its domain is [0, ∞), meaning it accepts any non-negative real number as input. The expression (x-8) inside the square root represents a horizontal shift. Specifically, it shifts the graph of the square root function 8 units to the right. This shift directly impacts the domain because it changes the range of x-values that will result in a non-negative expression inside the square root. The multiplication by 4 is a vertical stretch. It stretches the graph of the function vertically by a factor of 4. This affects the range of the function but does not alter the domain. In essence, the domain of y = 4√(x-8) is determined solely by the condition that the expression inside the square root, (x-8), must be non-negative. This function is a classic example of how transformations can affect the domain and range of a function. By identifying the function and understanding its components, we can set up the necessary inequality to solve for the domain. This careful analysis ensures we consider all relevant restrictions imposed by the function's structure, leading to an accurate determination of the set of valid input values.

To determine the domain of the function y = 4√(x-8), the key step is recognizing that the expression inside the square root must be non-negative. This is because the square root of a negative number is undefined within the realm of real numbers. Therefore, we set up the inequality x-8 ≥ 0. This inequality states that the expression (x-8) must be greater than or equal to zero. This is a critical condition that ensures the square root operation yields a real number. The inequality x-8 ≥ 0 captures the fundamental restriction imposed by the square root function. It translates the mathematical requirement into an algebraic statement that we can solve to find the domain. By setting up this inequality, we are essentially defining the boundaries of the input values that will produce valid outputs for the function. This step is crucial for accurately determining the domain. The inequality x-8 ≥ 0 is a linear inequality, which means it can be solved using basic algebraic techniques. Solving this inequality will give us the range of x-values that satisfy the condition, thereby defining the domain of the function. The solution to this inequality will provide the lower bound for the x-values. Any x-value less than this bound will result in a negative expression inside the square root, which is not allowed. The inequality x-8 ≥ 0 is a simple yet powerful tool for finding the domain of functions involving square roots. It encapsulates the essential requirement for the function to be defined in the real number system. This inequality is the foundation for the next step, which involves solving for x and expressing the domain in a suitable notation. In summary, setting up the inequality x-8 ≥ 0 is a pivotal step in finding the domain of the function y = 4√(x-8). It translates the mathematical restriction imposed by the square root into an algebraic condition that can be solved to determine the valid input values.

Having established the inequality x-8 ≥ 0, the next step is to solve for x. This involves isolating x on one side of the inequality to determine the range of values that satisfy the condition. To solve x-8 ≥ 0, we add 8 to both sides of the inequality. This is a standard algebraic manipulation that preserves the inequality. Adding 8 to both sides gives us: x-8 + 8 ≥ 0 + 8, which simplifies to x ≥ 8. This inequality, x ≥ 8, is the solution to the original inequality. It tells us that x must be greater than or equal to 8 for the function y = 4√(x-8) to produce a real number output. This is a clear and concise statement of the domain restriction. The solution x ≥ 8 is a crucial finding because it defines the set of all permissible input values for the function. Any value of x less than 8 will result in a negative number inside the square root, making the function undefined in the real number system. The inequality x ≥ 8 provides a lower bound for the domain. It means that 8 is the smallest value that x can take, and any value greater than 8 is also valid. This solution can be expressed in various forms, including interval notation and set notation. Understanding the solution x ≥ 8 is essential for graphing the function and interpreting its behavior. It informs us about the starting point of the graph along the x-axis and the direction in which the graph extends. The solution x ≥ 8 is a direct consequence of the restriction imposed by the square root function. It highlights the importance of considering the mathematical constraints when determining the domain of a function. In summary, solving the inequality x-8 ≥ 0 yields the solution x ≥ 8, which is the fundamental condition that defines the domain of the function y = 4√(x-8). This solution sets the stage for expressing the domain in a standard mathematical notation.

Once we've solved the inequality and found that x ≥ 8, the next step is to express this solution in interval notation. Interval notation is a standard way of representing sets of real numbers using intervals. It uses parentheses and brackets to indicate whether the endpoints are included or excluded from the set. The solution x ≥ 8 means that the domain includes all real numbers greater than or equal to 8. In interval notation, this is represented as [8, ∞). The square bracket '[' indicates that 8 is included in the domain. This is because the inequality x ≥ 8 includes the possibility that x is equal to 8. The parenthesis ')' indicates that infinity (∞) is not included in the domain. Infinity is not a number but rather a concept representing unboundedness, so it cannot be included as an endpoint. The interval [8, ∞) represents a continuous range of real numbers starting from 8 and extending indefinitely in the positive direction. This is a concise and clear way to express the domain of the function y = 4√(x-8). Interval notation is widely used in mathematics to represent intervals of real numbers. It provides a compact and unambiguous way to describe sets of numbers, making it a valuable tool for expressing domains and ranges of functions. The interval [8, ∞) is a specific example of a half-open interval, which means it includes one endpoint (8) but not the other (∞). This type of interval is common when dealing with functions that have restrictions on their domain or range. The use of interval notation helps to avoid ambiguity when describing sets of numbers. It clearly indicates whether the endpoints are included or excluded, which is crucial for understanding the behavior of the function. In summary, expressing the domain as [8, ∞) in interval notation provides a precise and standard way to represent the set of all valid input values for the function y = 4√(x-8). This notation captures the essence of the solution x ≥ 8 in a concise and widely understood format.

While interval notation is a common and efficient way to express the domain, there are alternative representations that can provide additional clarity or be more suitable in certain contexts. Two primary alternatives are set notation and graphical representation. Set notation uses curly braces to define a set. In this case, the domain of y = 4√(x-8) can be expressed in set notation as {x | x ∈ ℝ, x ≥ 8}. This notation reads as