Calculate F(2) + F(3) - F(1) Given F(x) = 5x - 2 A Comprehensive Guide
Given the function f(x) = 5x-2, calculate f(2) + f(3)-f(1).
In the realm of mathematics, understanding function evaluation is a cornerstone concept, particularly in algebra and calculus. A function, often denoted as f(x), is essentially a mathematical rule that assigns each input value (x) to a unique output value. The process of evaluating a function involves substituting a specific value for the variable (x) and then performing the operations indicated by the function's definition. This yields the corresponding output value.
1.1 The Essence of f(x)
The notation f(x), pronounced "f of x," represents the value of the function f at the input x. The variable x is the independent variable, and the output f(x) is the dependent variable. Evaluating f(x) for different values of x allows us to explore the function's behavior and its graphical representation.
1.2 The Significance of Function Evaluation
Function evaluation plays a vital role in various mathematical contexts, including solving equations, graphing functions, and modeling real-world phenomena. In calculus, function evaluation is crucial for determining limits, derivatives, and integrals. In practical applications, functions are used to model relationships between variables, such as the relationship between time and distance, or the relationship between price and demand. By evaluating these functions, we can make predictions and gain insights into the systems they represent.
1.3 Mastering Function Evaluation
To master function evaluation, it's crucial to understand the order of operations (PEMDAS/BODMAS) and to be meticulous in substituting values and performing calculations. Practice is key to developing fluency and confidence in evaluating functions of varying complexity.
In this article, we'll delve into the specific function f(x) = 5x - 2. This is a linear function, which means its graph is a straight line. Linear functions are fundamental in mathematics and have numerous applications in diverse fields.
2.1 The Structure of f(x) = 5x - 2
The function f(x) = 5x - 2 is composed of two terms: 5x and -2. The term 5x indicates that the input x is multiplied by 5, and the term -2 represents a constant value that is subtracted from the result. The coefficient 5 is the slope of the line, and the constant -2 is the y-intercept.
2.2 The Role of the Slope and Y-Intercept
The slope (5) determines the steepness and direction of the line. A positive slope indicates that the line rises as x increases, while a negative slope indicates that the line falls as x increases. The y-intercept (-2) is the point where the line crosses the y-axis. It represents the value of the function when x is 0.
2.3 Visualizing the Function
Visualizing the function f(x) = 5x - 2 can provide valuable insights into its behavior. The graph of this function is a straight line that passes through the point (0, -2) and has a slope of 5. This means that for every increase of 1 in x, the value of f(x) increases by 5.
To determine f(2) + f(3) - f(1), we must first evaluate the function f(x) = 5x - 2 at x = 2, x = 3, and x = 1. This involves substituting each value of x into the function's expression and performing the necessary calculations.
3.1 Evaluating f(2)
To find f(2), we substitute x = 2 into the function: f(2) = 5(2) - 2 = 10 - 2 = 8. Therefore, f(2) = 8.
3.2 Evaluating f(3)
Similarly, to find f(3), we substitute x = 3 into the function: f(3) = 5(3) - 2 = 15 - 2 = 13. Therefore, f(3) = 13.
3.3 Evaluating f(1)
To find f(1), we substitute x = 1 into the function: f(1) = 5(1) - 2 = 5 - 2 = 3. Therefore, f(1) = 3.
Now that we have determined the values of f(2), f(3), and f(1), we can proceed to calculate f(2) + f(3) - f(1). This involves substituting the calculated values into the expression and performing the arithmetic operations.
4.1 The Calculation
We have f(2) = 8, f(3) = 13, and f(1) = 3. Therefore, f(2) + f(3) - f(1) = 8 + 13 - 3 = 21 - 3 = 18.
4.2 The Result
The result of the calculation f(2) + f(3) - f(1) is 18.
In conclusion, this article has provided a comprehensive guide to understanding and performing function evaluation. We explored the essence of f(x), the significance of function evaluation in mathematics and real-world applications, and the importance of mastering this fundamental concept.
5.1 Key Takeaways
We deconstructed the function f(x) = 5x - 2, examining its structure, the roles of the slope and y-intercept, and the visualization of the function's graph. We then meticulously evaluated f(2), f(3), and f(1), demonstrating the process of substituting values and performing calculations. Finally, we calculated f(2) + f(3) - f(1), arriving at the solution of 18.
5.2 The Importance of Practice
Consistent practice is essential for solidifying your understanding of function evaluation and developing fluency in performing calculations. By working through various examples and exercises, you can build your confidence and accuracy in this crucial mathematical skill.
5.3 Applying Function Evaluation
Function evaluation is a powerful tool that extends far beyond the classroom. It is used extensively in fields such as physics, engineering, economics, and computer science to model and analyze real-world phenomena. By mastering function evaluation, you are equipping yourself with a valuable skill that will serve you well in your academic and professional endeavors.
In summary, understanding function evaluation is not just about performing calculations; it's about grasping the fundamental relationships between variables and gaining insights into the behavior of mathematical models. With dedication and practice, you can unlock the power of function evaluation and achieve success in your mathematical journey.