Understanding And Evaluating The Function F(x) = √(x) - 9

In mathematics, understanding functions is crucial for problem-solving and analytical thinking. This article delves into the function f(x) = √x - 9, exploring its behavior and evaluating it for specific values of x. We aim to provide a comprehensive guide that simplifies the function's application, making it accessible even for those new to mathematical functions. This function, involving a square root and subtraction, is a fundamental concept in algebra and calculus. By substituting different values for x, we can observe how the function behaves and the resulting outputs, which are essential for graphing and understanding the function's properties. This exploration is designed to enhance your understanding of functions and their practical applications in various mathematical scenarios. This guide will meticulously walk you through each step, ensuring clarity and comprehension, especially when dealing with negative values and the implications of square roots. By mastering this concept, you'll be better equipped to tackle more complex mathematical challenges.

The function f(x) = √x - 9 is a mathematical expression that takes an input x, calculates its square root, and then subtracts 9 from the result. The square root operation, denoted by √x, finds a value that, when multiplied by itself, equals x. For example, the square root of 25 is 5 because 5 * 5 = 25. However, it's crucial to remember that square roots of negative numbers are not real numbers, as there is no real number that, when multiplied by itself, results in a negative number. This aspect plays a significant role when evaluating the function for different values of x, particularly when dealing with negative inputs. The subtraction of 9 shifts the graph of the square root function downwards by 9 units, affecting its range and y-intercept. Understanding these individual components is key to grasping the overall behavior of the function. This section aims to dissect the function, providing a clear understanding of its components and how they interact. By breaking down the function, we can better predict its behavior and understand the outcomes for various inputs, which is a core skill in mathematical analysis.

To truly understand the function f(x) = √x - 9, we need to evaluate it for various values of x. This involves substituting specific numbers for x in the equation and calculating the corresponding output, f(x). This process helps us see how the function behaves for different inputs, particularly when dealing with negative numbers and the implications for real number outputs. Evaluating a function for different values is a fundamental skill in mathematics, as it allows us to plot graphs, identify key characteristics like intercepts and asymptotes, and understand the function's overall behavior. In this section, we'll methodically substitute each given value of x into the function and simplify the expression, paying close attention to the domain restrictions imposed by the square root. This hands-on approach will solidify your understanding of the function and its properties.

Evaluating f(-13)

Let's start by evaluating f(x) = √x - 9 when x = -13. Substituting -13 into the function, we get f(-13) = √(-13) - 9. Since we cannot take the square root of a negative number within the realm of real numbers, the result is not a real number. This is a critical concept in understanding the domain of square root functions, which are only defined for non-negative inputs. The appearance of a non-real number indicates that -13 is outside the function's domain. Recognizing these limitations is crucial in mathematical analysis and problem-solving. This step highlights the importance of understanding the constraints and conditions that apply to specific mathematical operations, particularly when dealing with roots and radicals.

Evaluating f(-1)

Next, let's evaluate f(x) = √x - 9 for x = -1. Substituting -1 into the function, we have f(-1) = √(-1) - 9. Similar to the previous case, the square root of a negative number is not a real number. Therefore, f(-1) is also not a real number. This reinforces the concept that square root functions are not defined for negative inputs within the real number system. This evaluation further illustrates the function's domain restrictions and the importance of considering them when working with mathematical expressions. Understanding when a function yields a non-real result is essential for accurately interpreting mathematical outcomes and avoiding errors.

Evaluating f(0)

Now, let's evaluate f(x) = √x - 9 when x = 0. Substituting 0 into the function, we get f(0) = √(0) - 9. The square root of 0 is 0, so the expression simplifies to f(0) = 0 - 9, which equals -9. Therefore, f(0) = -9. This result gives us a key point on the graph of the function: the y-intercept. The value of the function at x=0 is a fundamental characteristic that helps us understand the function's behavior and position on the coordinate plane. This calculation demonstrates how the function behaves at the origin and provides valuable information for sketching the function's graph.

Evaluating f(100)

Finally, let's evaluate f(x) = √x - 9 for x = 100. Substituting 100 into the function, we get f(100) = √(100) - 9. The square root of 100 is 10, so the expression becomes f(100) = 10 - 9, which equals 1. Therefore, f(100) = 1. This result gives us another significant point on the graph of the function, demonstrating how the function behaves for larger positive values of x. This calculation reinforces the process of evaluating the function for different inputs and provides another data point for understanding its overall behavior. The result highlights the function's increasing trend as x increases, within its defined domain.

To summarize our findings, we have evaluated the function f(x) = √x - 9 for four different values of x. For x = -13 and x = -1, the function resulted in not a real number due to the square root of a negative number. For x = 0, the function evaluated to -9, and for x = 100, the function evaluated to 1. These results provide a clear picture of the function's behavior, particularly its domain restrictions and the output values for specific inputs. This summary reinforces the key concepts explored in this article and provides a concise overview of the function's characteristics. The results highlight the importance of understanding domain restrictions and how they affect the evaluation of mathematical functions. This comprehensive overview serves as a valuable reference for future applications and problem-solving.

In conclusion, evaluating the function f(x) = √x - 9 for different values of x provides valuable insights into its behavior and characteristics. We've seen that the function is not defined for negative values of x within the real number system, highlighting the importance of considering domain restrictions. For x = 0, the function yields -9, and for x = 100, it yields 1. These results allow us to sketch the function's graph and understand its overall trend. This exploration reinforces the fundamental concepts of function evaluation and domain restrictions, which are essential in mathematics. Mastering these concepts enhances our ability to analyze and solve mathematical problems effectively. The detailed walkthrough provided in this article serves as a solid foundation for further exploration of functions and their applications in various mathematical contexts. By understanding how functions behave for different inputs, we can make informed decisions and predictions, which is a core skill in mathematical problem-solving.