Solving (f+g)(-1) For F(x) = X^2 + 5x And G(x) = 4x - 1 A Step-by-Step Guide
In mathematics, functions are fundamental building blocks that describe relationships between inputs and outputs. Often, we encounter scenarios where we need to combine functions or evaluate them at specific points. This article delves into the process of function addition and evaluation, specifically focusing on finding the value of (f+g)(-1) given the functions f(x) = x^2 + 5x and g(x) = 4x - 1. This problem exemplifies a common task in algebra and calculus, requiring a solid understanding of function notation and operations.
Defining Functions f(x) and g(x)
Let's first define the two functions involved in this problem:
- f(x) = x^2 + 5x
- g(x) = 4x - 1
The function f(x) is a quadratic function, characterized by the x² term. Its graph would be a parabola. The function g(x), on the other hand, is a linear function, representing a straight line when graphed. Our goal is to find the value of the combined function (f+g)(x) when x is equal to -1. Understanding the individual components of these functions is crucial before we proceed to the addition and evaluation steps. The quadratic function f(x) demonstrates a non-linear relationship, where the output changes at an increasing rate as x varies. Conversely, the linear function g(x) exhibits a constant rate of change, making it simpler to analyze. These distinct behaviors will influence the characteristics of the combined function (f+g)(x).
Understanding Function Addition: (f+g)(x)
The notation (f+g)(x) represents the sum of the two functions, f(x) and g(x). To find (f+g)(x), we simply add the expressions for f(x) and g(x) together. This means combining like terms to simplify the resulting expression. Function addition is a fundamental operation in function algebra, allowing us to create new functions by combining existing ones. In this case, by adding a quadratic function and a linear function, we will obtain another quadratic function, but with potentially different coefficients. The process of function addition can be visualized as vertically adding the graphs of the individual functions at each x-value. The resulting graph represents the combined function (f+g)(x). This concept extends to other function operations such as subtraction, multiplication, and division, each with its own set of rules and implications.
Calculating (f+g)(x)
Now, let's calculate (f+g)(x) using the given functions:
(f+g)(x) = f(x) + g(x)
Substitute the expressions for f(x) and g(x):
(f+g)(x) = (x^2 + 5x) + (4x - 1)
Combine like terms:
(f+g)(x) = x^2 + 5x + 4x - 1
(f+g)(x) = x^2 + 9x - 1
Therefore, the combined function (f+g)(x) is x^2 + 9x - 1. This resulting function is a quadratic function, inheriting the x² term from f(x). The coefficients have changed due to the addition of g(x), demonstrating how function addition modifies the original functions. Understanding how to perform this algebraic manipulation is crucial for evaluating the combined function at a specific point, which is our next step. The combined function (f+g)(x) represents a new mathematical entity with its own unique properties and graph. Analyzing this new function can provide insights into the relationship between the original functions and their combined behavior.
Evaluating (f+g)(-1)
To find (f+g)(-1), we substitute -1 for x in the expression for (f+g)(x):
(f+g)(-1) = (-1)^2 + 9(-1) - 1
Now, perform the arithmetic:
(f+g)(-1) = 1 - 9 - 1
(f+g)(-1) = -9
Therefore, (f+g)(-1) = -9. This result represents the y-value of the combined function (f+g)(x) when x is -1. Evaluating functions at specific points is a fundamental skill in mathematics, allowing us to understand the function's behavior at those points and make predictions. In this case, we found that the sum of the functions f(x) and g(x), evaluated at x = -1, is -9. This value can be interpreted graphically as the y-coordinate of the point on the graph of (f+g)(x) where x = -1. The process of substituting a value into a function and simplifying the expression is a core concept in algebra and calculus, used extensively in various applications.
Alternative Method: Evaluating f(-1) and g(-1) Separately
Alternatively, we could have evaluated f(-1) and g(-1) separately and then added the results:
First, find f(-1):
f(-1) = (-1)^2 + 5(-1)
f(-1) = 1 - 5
f(-1) = -4
Next, find g(-1):
g(-1) = 4(-1) - 1
g(-1) = -4 - 1
g(-1) = -5
Finally, add f(-1) and g(-1):
f(-1) + g(-1) = -4 + (-5)
f(-1) + g(-1) = -9
This method confirms our previous result: (f+g)(-1) = -9. This alternative approach highlights the flexibility of function operations. We can either add the functions first and then evaluate, or evaluate the functions separately and then add the results. Both methods are mathematically sound and will lead to the same answer. This flexibility is particularly useful in more complex scenarios where one method might be easier to apply than the other. Understanding the equivalence of these methods deepens our comprehension of function operations and their properties. It also reinforces the concept that functions can be manipulated algebraically, just like numbers.
Conclusion
In conclusion, we successfully found (f+g)(-1) to be -9, given the functions f(x) = x^2 + 5x and g(x) = 4x - 1. This involved understanding the concept of function addition, calculating the expression for (f+g)(x), and then evaluating it at x = -1. We also explored an alternative method of evaluating f(-1) and g(-1) separately and adding the results, demonstrating the flexibility of function operations. This problem illustrates the fundamental principles of working with functions, which are essential in various areas of mathematics and its applications. Mastering function addition and evaluation is a crucial step in developing a strong foundation in algebra and calculus. The ability to combine functions and evaluate them at specific points allows us to model and analyze a wide range of phenomena, from physical processes to economic trends. This foundational understanding is not only valuable for academic pursuits but also for problem-solving in real-world contexts.
Keywords: function addition, function evaluation, f(x) = x^2 + 5x, g(x) = 4x - 1, (f+g)(-1), quadratic function, linear function, algebra, mathematics.
