Evaluating Functions F(-1) + G(-1) A Step-by-Step Guide

In the world of mathematics, functions play a crucial role in modeling relationships between variables. Understanding how to evaluate functions and perform operations on them is fundamental to various mathematical concepts. In this comprehensive guide, we will delve into the process of evaluating functions and calculating their sum, using the specific example of f(x) = 2x² + 3x - 1 and g(x) = -x. By following the step-by-step instructions and explanations, you will gain a solid understanding of these concepts and be able to apply them to other similar problems. So, let's embark on this mathematical journey together and unlock the secrets of function evaluation and addition.

Understanding Functions and Function Notation

Before we dive into the evaluation process, let's first understand what functions are and how they are represented mathematically. In simple terms, a function is a mathematical rule that assigns a unique output value to each input value. We can think of a function as a machine that takes an input, processes it according to a specific rule, and produces an output. For example, consider the function f(x) = 2x² + 3x - 1. This function takes an input value 'x', squares it, multiplies it by 2, adds 3 times the input value, and then subtracts 1 to produce the output value. The notation f(x) represents the output value of the function f when the input value is 'x'. The variable 'x' is called the independent variable or the input variable, while the output value f(x) is called the dependent variable or the output variable. Understanding the concept of functions and function notation is essential for evaluating functions and performing operations on them. Now that we have a clear understanding of functions, let's move on to the next step: evaluating functions for specific input values.

Evaluating f(x) at x = -1

To evaluate a function for a specific input value, we simply substitute the input value into the function's expression and simplify. In this case, we want to find the value of f(x) when x = -1. So, we substitute -1 for x in the expression for f(x): f(-1) = 2(-1)² + 3(-1) - 1. Now, we follow the order of operations (PEMDAS/BODMAS) to simplify the expression. First, we evaluate the exponent: (-1)² = 1. Then, we perform the multiplications: 2(1) = 2 and 3(-1) = -3. Now we put all the values back into the function and perform the additions and subtractions from left to right: f(-1) = 2 - 3 - 1. Finally, 2 - 3 = -1, and -1 - 1 = -2. Therefore, f(-1) = -2. This means that when the input value is -1, the output value of the function f(x) is -2. Evaluating functions at specific input values is a fundamental skill in mathematics, and it forms the basis for many other mathematical concepts and applications. In the next section, we will evaluate the function g(x) for the same input value, x = -1.

Evaluating g(x) at x = -1

Now that we have evaluated f(x) at x = -1, let's evaluate g(x) at x = -1. The function g(x) is defined as g(x) = -x. This means that the output value of the function g for any input value 'x' is simply the negative of that input value. To find g(-1), we substitute -1 for x in the expression for g(x): g(-1) = -(-1). The negative of a negative number is the positive version of that number, so -(-1) = 1. Therefore, g(-1) = 1. This means that when the input value is -1, the output value of the function g(x) is 1. Evaluating functions like g(x) is straightforward, as it involves a simple operation. However, it's important to understand the concept of function evaluation and how to apply it to different types of functions. In the next section, we will combine the results we obtained for f(-1) and g(-1) to calculate their sum.

Calculating f(-1) + g(-1)

In this section, we will calculate the sum of f(-1) and g(-1). We have already found that f(-1) = -2 and g(-1) = 1. To find the sum, we simply add these two values together: f(-1) + g(-1) = -2 + 1. Adding a negative number and a positive number is the same as subtracting the smaller number from the larger number and using the sign of the larger number. In this case, the larger number is 2 (in absolute value), and it has a negative sign. So, we subtract 1 from 2, which gives us 1, and then we use the negative sign, which gives us -1. Therefore, f(-1) + g(-1) = -1. This means that the sum of the output values of the functions f(x) and g(x) when x = -1 is -1. Calculating the sum of function values is a common operation in mathematics, and it allows us to combine the behaviors of different functions. In the next section, we will summarize the steps we took to solve this problem and highlight the key concepts involved.

Summary and Key Takeaways

In this guide, we have explored the process of evaluating functions and calculating their sum. We started by understanding the concept of functions and function notation, which are fundamental to working with functions. We then evaluated the functions f(x) = 2x² + 3x - 1 and g(x) = -x at x = -1. To evaluate f(-1), we substituted -1 for x in the expression for f(x) and simplified, following the order of operations. To evaluate g(-1), we substituted -1 for x in the expression for g(x), which simply involved taking the negative of -1. Finally, we calculated the sum of f(-1) and g(-1) by adding the values we obtained for each function. The key takeaways from this guide are: * Functions are mathematical rules that assign unique output values to input values. * Function notation, such as f(x), represents the output value of a function f for an input value x. * To evaluate a function for a specific input value, substitute the input value into the function's expression and simplify. * The order of operations (PEMDAS/BODMAS) should be followed when simplifying expressions. * The sum of function values can be calculated by adding the output values of the functions for a specific input value. By understanding these concepts and practicing function evaluation and addition, you will build a strong foundation for further mathematical studies.

By mastering the techniques outlined in this guide, you'll be well-equipped to tackle a wide range of mathematical problems involving functions. Remember, practice is key to solidifying your understanding, so don't hesitate to work through additional examples and exercises. With dedication and perseverance, you'll unlock the power of functions and their applications in mathematics and beyond.

Keywords: function evaluation, function notation, sum of functions, mathematical functions, evaluating expressions