Evaluating F(x) = -x - 7 For Given X Values A Step By Step Guide

In the realm of mathematics, functions play a pivotal role in describing relationships between variables. Understanding functions is crucial for various mathematical concepts, including calculus, algebra, and data analysis. At its core, a function is like a machine: you input a value, and it spits out another value based on a specific rule. This article delves into the world of function evaluation, focusing on a linear function, $f(x) = -x - 7$. We'll explore how to determine the output or $y$-value for given $x$-values, a fundamental skill in mathematics.

Function evaluation is the process of finding the output of a function for a given input. It involves substituting the input value into the function's expression and simplifying. This process allows us to understand how the function transforms different inputs and reveals the nature of the relationship it represents. In this article, we will evaluate the function $f(x) = -x - 7$ for a set of given $x$-values, demonstrating the step-by-step process and highlighting key concepts. Mastering function evaluation is essential for solving mathematical problems, modeling real-world phenomena, and gaining a deeper understanding of mathematical relationships.

The function we're working with today is $f(x) = -x - 7$. This is a linear function, meaning that when graphed, it forms a straight line. The $f(x)$ notation represents the output or $y$-value of the function for a given input $x$. The function's expression, $-x - 7$, tells us exactly how to transform the input value. In this case, it means we take the negative of the input $x$ and then subtract 7. For instance, if we input $x = 2$, we would calculate $f(2) = -2 - 7 = -9$. Understanding the structure of a function is crucial for evaluating it correctly. Linear functions are particularly important because they model many real-world relationships, such as the relationship between time and distance at a constant speed or the relationship between the number of items purchased and the total cost. The function $f(x) = -x - 7$ is a simple yet powerful example of how mathematical expressions can represent and describe patterns.

Now, let's get to the core of the problem: determining the $y$-values for specific $x$-values. We have a table with three $x$-values: -4, 3, and 7. Our task is to find the corresponding $y$-values by plugging each $x$-value into the function $f(x) = -x - 7$. This process involves careful substitution and arithmetic. We will systematically go through each $x$-value, demonstrating the calculation step-by-step. Understanding how to evaluate a function for different inputs is a fundamental skill in mathematics. It allows us to map the behavior of the function and gain insights into its properties. This process is not only essential for solving mathematical problems but also for applying mathematical models to real-world scenarios. So, let's begin our journey of function evaluation and uncover the hidden outputs of $f(x) = -x - 7$.

Case 1: x = -4

When $x = -4$, we substitute this value into our function: $f(-4) = -(-4) - 7$. Notice the double negative – this is a crucial detail! The negative of a negative number is positive, so $-(-4)$ becomes 4. Now we have $f(-4) = 4 - 7$. Subtracting 7 from 4 gives us -3. Therefore, when $x = -4$, $y = -3$. This illustrates how the function transforms a negative input into a negative output. The careful handling of signs is essential in mathematical calculations, and this example emphasizes its importance. By understanding how to substitute negative values correctly, we can accurately evaluate functions and interpret their behavior across the number line. This skill is not only applicable to linear functions but also to more complex functions encountered in higher-level mathematics.

Case 2: x = 3

Next, we consider $x = 3$. Substituting this into our function gives us $f(3) = -(3) - 7$. This is a straightforward substitution. We have $f(3) = -3 - 7$. Adding two negative numbers results in a larger negative number, so $-3 - 7 = -10$. Thus, when $x = 3$, $y = -10$. This demonstrates how the function produces a negative output for a positive input. The consistent application of arithmetic rules is vital for accurate function evaluation. This example reinforces the concept of adding negative numbers and highlights the impact of the function's expression on the output. By mastering these basic arithmetic operations within the context of function evaluation, we build a strong foundation for more advanced mathematical concepts.

Case 3: x = 7

Finally, let's evaluate the function for $x = 7$. Substituting this value into the function gives us $f(7) = -(7) - 7$. Again, this is a direct substitution. We have $f(7) = -7 - 7$. Adding two negative numbers gives us $-14$. So, when $x = 7$, $y = -14$. This further illustrates the function's tendency to produce negative outputs. The repetition of the process with different inputs reinforces the method of function evaluation. This example serves as a final check of our understanding and emphasizes the consistency of the mathematical rules involved. By confidently evaluating the function for a range of inputs, we solidify our grasp of the function's behavior and its relationship to the coordinate plane.

Now that we've calculated the $y$-values for each $x$-value, we can complete the table:

$ \begin{array}{c c} x & y
-4 & -3 \ 3 & -10 \ 7 & -14 \ \end{array} $

This table provides a clear representation of the function's behavior for the given $x$-values. Each row shows an input-output pair, allowing us to visualize the relationship described by the function. The table format is a valuable tool for organizing and presenting data, especially in mathematical contexts. It facilitates pattern recognition and provides a concise summary of the function's transformations. By completing the table, we not only solve the problem but also gain a deeper understanding of the function's properties and its role in mapping inputs to outputs. This skill is crucial for various mathematical applications, including graphing functions, solving equations, and analyzing data.

In this article, we successfully evaluated the function $f(x) = -x - 7$ for various $x$-values. We demonstrated the process of substituting the $x$-values into the function and simplifying to find the corresponding $y$-values. This exercise highlights the fundamental concept of function evaluation and its importance in mathematics. By understanding how to evaluate functions, we can analyze their behavior, solve equations, and model real-world phenomena. The function $f(x) = -x - 7$, being a linear function, provides a clear and accessible example for grasping these concepts. The step-by-step approach we followed can be applied to a wide range of functions, regardless of their complexity. Function evaluation is a cornerstone of mathematical understanding, and mastering this skill opens doors to more advanced topics and applications. As we continue our mathematical journey, the ability to confidently evaluate functions will serve as a valuable asset.

• Function Evaluation
• Linear Function
• f(x) = -x - 7
• Evaluating Functions
• Mathematics
• Algebra
• Input-Output
• Y-value
• X-value
• Table
• Function
• Solving for Y