Finding The Function Value Of F(x) = √(x-5) At X = 30
In mathematics, functions are fundamental tools for describing relationships between variables. Evaluating a function at a specific point involves substituting that value into the function's expression and simplifying. This process allows us to determine the output of the function for a given input. In this article, we will delve into the process of finding the function value for the function f(x) = √(x-5) when x = 30. This involves a straightforward substitution and simplification process, which we will break down step by step. Understanding how to evaluate functions is crucial in various areas of mathematics, including calculus, algebra, and mathematical analysis. The ability to accurately find function values is essential for graphing functions, solving equations, and analyzing real-world scenarios modeled by mathematical functions.
Before we dive into the specific problem, let's briefly discuss what a function is. A function is a mathematical relationship that maps each input value from a set (called the domain) to a unique output value in another set (called the range). Functions are typically denoted using a letter, such as f, g, or h, followed by the input variable in parentheses, like f(x). The expression on the right side of the equals sign defines the rule or formula for how the function operates on the input. For instance, in our case, the function f(x) = √(x-5) tells us to take the input value x, subtract 5 from it, and then take the square root of the result. This definition is crucial for understanding how to evaluate the function for different values of x. Functions are a cornerstone of mathematical modeling and are used extensively in various scientific and engineering disciplines. They allow us to represent complex relationships in a concise and manageable way, making them indispensable tools for problem-solving and analysis.
Our specific function is f(x) = √(x-5). This is a square root function, which means it involves taking the square root of an expression. The expression inside the square root, known as the radicand, is x - 5. It's important to note that the square root function has a domain restriction. Since we cannot take the square root of a negative number (in the realm of real numbers), the radicand x - 5 must be greater than or equal to zero. This gives us the inequality x - 5 ≥ 0, which simplifies to x ≥ 5. Therefore, the domain of this function is all real numbers greater than or equal to 5. This is a crucial consideration when evaluating the function; we must ensure that the input value x is within the function's domain. Understanding the domain of a function is essential for interpreting its behavior and ensuring that the results obtained are meaningful. Square root functions are widely used in mathematics and physics to model various phenomena, such as distances, speeds, and physical constraints.
Now, let's evaluate the function at x = 30. This means we need to substitute 30 for x in the expression for f(x). So, we have:
f(30) = √(30 - 5)
The next step is to simplify the expression inside the square root:
30 - 5 = 25
Now, we have:
f(30) = √25
Finally, we take the square root of 25:
√25 = 5
Therefore, the value of the function f(x) = √(x-5) at x = 30 is 5. This straightforward substitution and simplification process demonstrates the fundamental principle of function evaluation. By replacing the variable x with the specific value 30, we have successfully determined the output of the function. This skill is crucial for various mathematical tasks, such as graphing functions, solving equations, and analyzing data.
Let's recap the step-by-step solution to ensure clarity:
- Write down the function: f(x) = √(x-5)
- Substitute x = 30: f(30) = √(30 - 5)
- Simplify the expression inside the square root: 30 - 5 = 25
- Rewrite the expression: f(30) = √25
- Evaluate the square root: √25 = 5
- State the result: f(30) = 5
This methodical approach highlights the importance of following a clear and organized process when evaluating functions. Each step builds upon the previous one, ensuring that the final result is accurate. This systematic approach is particularly useful when dealing with more complex functions or when evaluating functions at multiple points. By breaking down the problem into manageable steps, we can minimize errors and gain a deeper understanding of the function's behavior. This step-by-step method can be applied to a wide range of function evaluation problems, making it a valuable tool in mathematical problem-solving.
In conclusion, we have successfully found the function value of f(x) = √(x-5) at x = 30, which is 5. This exercise demonstrates the fundamental process of function evaluation: substituting the input value into the function's expression and simplifying. Understanding how to evaluate functions is crucial for various mathematical applications and problem-solving scenarios. The step-by-step approach we used ensures accuracy and clarity, which are essential when working with mathematical functions. This skill is a building block for more advanced mathematical concepts and is widely applicable in various fields that rely on mathematical modeling and analysis. Mastering function evaluation is a key step in developing a strong foundation in mathematics and its applications.
Function value, square root function, function evaluation, domain of a function, substitution, simplification, mathematical analysis, graphing functions, solving equations, mathematical modeling.
