Find The Domain Of The Function U(x) = √(-9x + 18) Explained

Determining the domain of a function is a fundamental concept in mathematics, particularly in precalculus and calculus. The domain 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 encountering any mathematical errors, such as division by zero or taking the square root of a negative number. This article will delve into the process of finding the domain of the function $u(x) = \sqrt{-9x + 18}$, providing a step-by-step explanation and emphasizing the importance of interval notation in expressing the solution. Understanding the domain is crucial for analyzing the behavior of functions, graphing them accurately, and applying them to real-world problems. The domain helps us to define the boundaries within which the function operates meaningfully and provides valuable insights into the function's characteristics. By mastering the techniques for finding the domain, you'll be better equipped to tackle more advanced mathematical concepts and applications.

Understanding the Domain of a Function

To effectively find the domain, it's essential to grasp the underlying concept. The domain of a function is the set of all input values (x-values) for which the function produces a real and defined output. Certain mathematical operations impose restrictions on the domain. For example, division by zero is undefined, so any x-value that would cause the denominator of a function to be zero must be excluded from the domain. Similarly, the square root of a negative number is not a real number, so any x-values that would result in a negative value under the square root must also be excluded. These restrictions are critical to consider when determining the domain of a function, ensuring that the function operates within valid mathematical boundaries. Failing to account for these restrictions can lead to incorrect results and a misunderstanding of the function's behavior. Therefore, a thorough understanding of these limitations is crucial for accurate analysis and application of mathematical functions. The concept of the domain is not just a theoretical exercise; it has practical implications in various fields, including physics, engineering, and economics, where functions are used to model real-world phenomena. In these applications, the domain often represents physical constraints or limitations on the variables involved. For instance, in a function that models the height of a projectile, the domain might be restricted to non-negative values of time. In economics, the domain of a cost function might be limited to positive quantities of goods produced. Therefore, understanding the domain is essential for interpreting and applying mathematical models in real-world contexts.

Identifying Restrictions: Square Root Functions

In the given function, $u(x) = \sqrt{-9x + 18}$, the primary restriction arises from the square root. The expression inside the square root, known as the radicand, must be greater than or equal to zero to ensure a real-valued output. This is because the square root of a negative number is not defined within the realm of real numbers. Therefore, the key to finding the domain of this function lies in identifying the values of x that make the radicand, $-9x + 18$, non-negative. This restriction forms the foundation for determining the valid input values for the function. Ignoring this restriction would lead to imaginary or complex outputs, which are not within the scope of the real-valued domain we are seeking. The principle of a non-negative radicand applies to all square root functions, and it's a fundamental rule to remember when dealing with such functions. This restriction is not just a mathematical technicality; it has practical implications in various applications. For example, if this function were modeling a physical quantity that cannot be negative, the restriction would ensure that the model remains physically meaningful. Similarly, in computer science, square root functions are used in various algorithms, and ensuring a non-negative radicand is crucial for avoiding errors and ensuring the algorithm's stability. Therefore, understanding this restriction is essential for both theoretical and practical applications of square root functions. Furthermore, this restriction is a specific instance of a broader principle: even-indexed roots (such as fourth roots, sixth roots, etc.) of negative numbers are not real numbers. The same principle applies to these functions, and the radicand must be non-negative to ensure a real-valued output.

Setting Up the Inequality

To find the domain, we need to ensure that the radicand, $-9x + 18$, is greater than or equal to zero. This translates into the following inequality: $-9x + 18 \ge 0$. This inequality mathematically expresses the condition that the expression under the square root must be non-negative. Solving this inequality will give us the range of x-values that satisfy this condition, and thus, the domain of the function. The inequality represents a constraint on the input variable 'x', ensuring that the function's output remains within the set of real numbers. Setting up this inequality is a critical step in the process, as it transforms the problem of finding the domain into a problem of solving an algebraic inequality. This approach is a standard technique for finding the domains of functions involving square roots and other even-indexed roots. The inequality $-9x + 18 \ge 0$ is a linear inequality, which means it involves a linear expression in 'x'. Linear inequalities are relatively straightforward to solve, and the solution will typically be an interval or a union of intervals. However, it's important to pay attention to the direction of the inequality sign when multiplying or dividing by a negative number, as this will affect the solution. The inequality $-9x + 18 \ge 0$ is a fundamental representation of the domain restriction, and its solution will provide the precise set of x-values that can be used as inputs for the function. This inequality is not just a mathematical abstraction; it has practical implications in various applications. For example, if this function were modeling a physical quantity, the inequality would ensure that the model remains physically meaningful.

Solving the Inequality

Now, let's solve the inequality $-9x + 18 \ge 0$. First, we isolate the term with 'x' by subtracting 18 from both sides: $-9x \ge -18$. Next, we divide both sides by -9. Crucially, when dividing or multiplying an inequality by a negative number, we must reverse the direction of the inequality sign. This gives us $x \le 2$. This is a critical step in solving inequalities, and forgetting to reverse the inequality sign can lead to an incorrect solution. The reason for reversing the sign is that multiplying or dividing by a negative number changes the order of the numbers on the number line. For example, while 2 is greater than 1, -2 is less than -1. Therefore, when we divide by -9, we are essentially changing the sign of the numbers, and we must reverse the inequality to maintain the correct relationship. The solution $x \le 2$ represents all the x-values that are less than or equal to 2. This means that any x-value greater than 2 would make the radicand negative, resulting in an imaginary output. The solution $x \le 2$ is a key result in determining the domain of the function. It provides a clear and concise representation of the allowed input values. However, to fully understand and communicate the domain, it's essential to express this solution in interval notation, which is a standard way of representing sets of numbers.

Expressing the Solution in Interval Notation

The solution $x \le 2$ can be expressed in interval notation as $(-\infty, 2]$. Interval notation is a convenient and standardized way to represent a set of real numbers. The parenthesis '(' indicates that the endpoint is not included in the interval, while the bracket '[' indicates that the endpoint is included. In this case, $(-\infty, 2]$ represents all real numbers less than or equal to 2. The symbol $-\infty$ represents negative infinity, indicating that the interval extends indefinitely in the negative direction. The bracket on the 2 indicates that 2 is included in the interval, which is consistent with the inequality $x \le 2$. Interval notation is a powerful tool for representing domains and ranges of functions, and it's widely used in mathematics. It provides a clear and concise way to communicate the set of allowed input values or output values. Understanding interval notation is essential for working with functions and their properties. It allows us to express complex sets of numbers in a compact and unambiguous way. Furthermore, interval notation is not just a mathematical convention; it has practical applications in various fields. For example, in statistics, interval notation is used to represent confidence intervals, which provide a range of values within which a population parameter is likely to lie. In computer science, interval notation is used to represent ranges of values for variables and data structures. Therefore, mastering interval notation is a valuable skill for anyone working with quantitative data.

Final Answer: The Domain of $u(x)$

Therefore, the domain of the function $u(x) = \sqrt{-9x + 18}$ is $(-\infty, 2]$. This means that the function is defined for all real numbers less than or equal to 2. Any value greater than 2 would result in a negative value under the square root, making the function undefined in the real number system. The domain provides a clear boundary for the input values, ensuring that the function operates within a valid mathematical context. Understanding the domain is crucial for analyzing the function's behavior, graphing it accurately, and applying it to real-world problems. The domain is not just a theoretical concept; it has practical implications in various fields where functions are used to model real-world phenomena. For example, if this function were modeling a physical quantity, the domain would represent the range of input values for which the model is physically meaningful. Similarly, in computer science, the domain would define the valid input range for a function or algorithm. Therefore, determining the domain is an essential step in the analysis and application of any function. Furthermore, the domain is closely related to other important concepts, such as the range, intercepts, and asymptotes of a function. Understanding the domain helps us to understand the overall behavior of the function and its relationship to the input and output values. Therefore, a thorough understanding of the domain is crucial for a comprehensive understanding of functions.