Evaluating F(-1) + G(-1) Where F(x) = 2x² + 3x - 1 And G(x) = -x
In the realm of mathematics, functions serve as fundamental building blocks for modeling relationships and processes. They elegantly describe how an input value transforms into an output value, providing a concise and powerful way to represent complex phenomena. This article delves into the evaluation of functions, specifically focusing on the scenario where we have two functions, f(x) and g(x), and we aim to determine the sum of their values at a particular input, in this case, x = -1. We will explore the step-by-step process of evaluating each function individually and then combining the results to arrive at the final answer. This exploration will not only solidify your understanding of function evaluation but also highlight the importance of attention to detail and the application of basic arithmetic operations in mathematical problem-solving. Let's embark on this journey of mathematical discovery and unravel the intricacies of function evaluation.
Before diving into the evaluation process, let's first gain a clear understanding of the functions we're dealing with. We have two functions at hand:
- f(x) = 2x² + 3x - 1: This is a quadratic function, characterized by the presence of a term with x raised to the power of 2. Quadratic functions are known for their parabolic shapes when graphed and play a crucial role in various mathematical and real-world applications.
- g(x) = -x: This is a linear function, which represents a straight line when graphed. Linear functions are the simplest type of functions and are widely used to model direct relationships between variables.
Each function takes an input value (x) and applies a specific set of operations to it to produce an output value. The function f(x) involves squaring the input, multiplying it by 2, adding 3 times the input, and then subtracting 1. The function g(x) simply negates the input value. Understanding these operations is essential for accurately evaluating the functions at a given input. In the next sections, we'll explore how to substitute a specific value for x and perform the necessary calculations to determine the output of each function.
Evaluating f(x) at x = -1
The core concept of function evaluation lies in substituting a specific value for the variable x within the function's expression. In our case, we want to find the value of f(x) when x is -1, which is denoted as f(-1). To do this, we replace every instance of x in the expression for f(x) with -1. This gives us:
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
Next, we perform the multiplications:
2(1) = 2 3(-1) = -3
Now, we substitute these results back into the expression:
f(-1) = 2 - 3 - 1
Finally, we perform the additions and subtractions from left to right:
f(-1) = -1 - 1 f(-1) = -2
Therefore, the value of the function f(x) when x is -1 is -2. This means that the input -1 is mapped to the output -2 by the function f. This step-by-step process ensures that we accurately calculate the function's value at the given input. The careful application of the order of operations is crucial to avoid errors and arrive at the correct result. In the next section, we will apply a similar process to evaluate the function g(x) at x = -1.
Evaluating g(x) at x = -1
Now, let's shift our focus to the function g(x) = -x. Our goal is to find the value of g(x) when x is -1, which is written as g(-1). Similar to our approach with f(x), we substitute -1 for x in the expression for g(x). This gives us:
g(-1) = -(-1)
The key here is to recognize that we are negating a negative number. The negative of a negative number is its positive counterpart. Therefore:
g(-1) = 1
This simple yet important step highlights the role of signs in mathematical operations. The function g(x), in this case, acts as a sign inverter, transforming a negative input into a positive output. The value of the function g(x) when x is -1 is 1. This means that the input -1 is mapped to the output 1 by the function g. This straightforward evaluation reinforces the concept of function evaluation as a substitution and simplification process. Having evaluated both f(-1) and g(-1), we are now ready to combine these results to find their sum, which will provide the final answer to our problem. In the next section, we will perform this addition and conclude our exploration.
Finding f(-1) + g(-1)
Having diligently evaluated f(-1) and g(-1), we now stand at the final step of our journey: finding the sum of these two values. We previously determined that:
- f(-1) = -2
- g(-1) = 1
To find f(-1) + g(-1), we simply add these two values together:
f(-1) + g(-1) = -2 + 1
Performing the addition, we get:
f(-1) + g(-1) = -1
Therefore, the sum of f(-1) and g(-1) is -1. This result encapsulates the entire process we've undertaken, from understanding the functions to evaluating them at a specific input and finally combining the results. This final calculation underscores the interconnectedness of mathematical operations and the importance of accurate execution at each step. The result, -1, represents the combined output of the two functions f and g when evaluated at x = -1. This value provides a concise answer to the problem and demonstrates the power of function evaluation in determining the behavior of mathematical models.
In this exploration, we've meticulously dissected the process of evaluating functions and finding the sum of their values at a specific input. We started by understanding the definitions of the functions f(x) = 2x² + 3x - 1 and g(x) = -x. We then embarked on the evaluation journey, first finding f(-1) by substituting -1 for x in the expression for f(x) and simplifying using the order of operations. This yielded f(-1) = -2. Next, we evaluated g(-1) by substituting -1 for x in the expression for g(x), which resulted in g(-1) = 1. Finally, we combined these results by adding f(-1) and g(-1), arriving at the answer f(-1) + g(-1) = -1. This comprehensive process highlights the significance of careful substitution, adherence to the order of operations, and accurate arithmetic calculations in mathematical problem-solving.
This exercise not only reinforces the fundamental concepts of function evaluation but also demonstrates the practical application of these concepts in determining the behavior of mathematical models. The ability to evaluate functions is a cornerstone of mathematical understanding and is essential for tackling more advanced topics in calculus, analysis, and other areas of mathematics. By mastering these foundational skills, we empower ourselves to explore the intricate world of mathematical relationships and their applications in various fields of science, engineering, and beyond.
