Finding F(5) Given F(x) = X² - 2x + 4 A Step-by-Step Solution
In mathematics, functions are a fundamental concept. They describe a relationship between inputs and outputs. Evaluating a function at a specific point is a crucial skill. This article provides a detailed walkthrough of how to find the value of a function at a given point, using the example of the quadratic function F(x) = x² - 2x + 4. We will specifically focus on finding F(5), which means determining the output of the function when the input is 5. This process involves substituting the input value into the function's expression and simplifying the resulting expression using the order of operations. Understanding how to evaluate functions is essential for various mathematical applications, including graphing, solving equations, and modeling real-world phenomena. This concept forms the basis for more advanced topics in algebra and calculus, making it a cornerstone of mathematical literacy.
Understanding Functions
Before diving into the specific problem, let's solidify our understanding of functions. A function is like a machine that takes an input, processes it according to a specific rule, and produces an output. The function is usually denoted by a letter, such as F, g, or h, and the input is represented by a variable, often x. The rule that the function applies to the input is expressed as a mathematical expression. For instance, in our example, the function F is defined by the expression x² - 2x + 4. This means that when we input a value for x, the function will square it, subtract twice the value of x, and then add 4. The output of the function is denoted by F(x), which is read as "F of x." The key to working with functions is to understand that F(x) represents the result of applying the function's rule to the input x. To evaluate a function at a specific point, we simply replace the variable x in the function's expression with the given value and simplify. This process is fundamental to understanding the behavior of functions and their applications in various fields of mathematics and beyond. The ability to accurately evaluate functions is a building block for more advanced mathematical concepts, such as calculus and differential equations. It also allows us to model real-world phenomena and make predictions based on mathematical relationships. For example, in physics, functions can describe the motion of objects, and in economics, they can model market behavior. Therefore, mastering the evaluation of functions is crucial for anyone pursuing studies or careers in STEM fields.
The Function F(x) = x² - 2x + 4
In this particular problem, we are given the function F(x) = x² - 2x + 4. This is a quadratic function, which means it has the form of a polynomial of degree two. The x² term is the highest power of x, making it a quadratic. Quadratic functions are characterized by their parabolic shape when graphed. The coefficients of the terms determine the shape and position of the parabola. In our function, the coefficient of x² is 1, the coefficient of x is -2, and the constant term is 4. These coefficients play a crucial role in determining the properties of the parabola, such as its vertex (the minimum or maximum point) and its axis of symmetry. The general form of a quadratic function is ax² + bx + c, where a, b, and c are constants. In our case, a = 1, b = -2, and c = 4. Understanding the coefficients allows us to analyze the behavior of the function and predict its output for different input values. For example, the positive coefficient of x² indicates that the parabola opens upwards, meaning it has a minimum value. The negative coefficient of x shifts the parabola horizontally, and the constant term shifts it vertically. Quadratic functions have numerous applications in real-world scenarios. They can be used to model projectile motion, the trajectory of a ball thrown in the air, or the shape of a suspension bridge cable. They also appear in optimization problems, where we seek to find the maximum or minimum value of a quantity, such as the profit of a business or the area of a garden. Therefore, understanding quadratic functions and how to work with them is essential in various fields, including physics, engineering, economics, and computer science.
The Task: Finding F(5)
Our task is to find F(5). This notation means we need to evaluate the function F when the input x is equal to 5. In other words, we need to substitute 5 for x in the expression that defines the function and then simplify the resulting expression. This process is a direct application of the definition of a function: we are simply applying the function's rule to the specific input value. The notation F(5) represents the output of the function when the input is 5. It is important to distinguish this from other mathematical notations, such as multiplication or composition of functions. F(5) is not the same as F multiplied by 5; it is the value of the function F at the point x = 5. Finding F(5) is a specific instance of evaluating the function, but the process is the same for any other input value. For example, to find F(0), we would substitute 0 for x in the expression. To find F(-2), we would substitute -2 for x. The key is to consistently apply the function's rule to the given input value. Evaluating functions at specific points is a fundamental skill in mathematics, and it is used extensively in various applications. For example, in calculus, we often need to evaluate functions to find limits, derivatives, and integrals. In statistics, we may need to evaluate probability density functions to calculate probabilities. In computer science, functions are used extensively in programming, and evaluating them is a core operation in algorithm execution. Therefore, mastering the process of finding F(5) and other function evaluations is crucial for success in various fields.
Step-by-Step Solution: Substituting x = 5
To find F(5), we begin by substituting x = 5 into the function's expression: F(x) = x² - 2x + 4. Replacing x with 5, we get:
F(5) = (5)² - 2(5) + 4
This substitution is the crucial first step in evaluating the function at the given point. It transforms the symbolic expression of the function into a numerical expression that we can simplify. It is important to perform the substitution carefully, ensuring that each instance of x is replaced with the value 5. Errors in substitution can lead to incorrect results. The parentheses around the 5 in (5)² and 2(5) are important for clarity. They indicate that we are squaring the entire value 5 and multiplying 2 by the value 5. Without the parentheses, the expression could be misinterpreted. The next step in the solution involves simplifying this numerical expression using the order of operations. This order, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which we perform the operations. Following the order of operations is essential for obtaining the correct result. In this case, we will first evaluate the exponent, then perform the multiplication, and finally carry out the addition and subtraction. The result of this substitution sets the stage for the subsequent calculations that will lead us to the final value of F(5). This process of substitution and simplification is fundamental to evaluating functions and understanding their behavior.
Step-by-Step Solution: Simplifying the Expression
Now, we simplify the expression F(5) = (5)² - 2(5) + 4 following the order of operations (PEMDAS). First, we evaluate the exponent: (5)² = 5 * 5 = 25. So, the expression becomes:
F(5) = 25 - 2(5) + 4
Next, we perform the multiplication: 2(5) = 10. The expression now is:
F(5) = 25 - 10 + 4
Finally, we perform the subtraction and addition from left to right: 25 - 10 = 15, and then 15 + 4 = 19. Therefore,
F(5) = 19
This step-by-step simplification demonstrates the importance of following the order of operations. Evaluating the exponent first, then the multiplication, and finally the addition and subtraction ensures that we arrive at the correct result. Ignoring the order of operations can lead to a different and incorrect answer. Each step in the simplification process is a direct application of basic arithmetic operations. The squaring of 5 is a fundamental multiplication operation. The multiplication of 2 and 5 is another basic arithmetic operation. The subtraction and addition are also fundamental operations that we use extensively in mathematics. The final result, F(5) = 19, tells us the output of the function F when the input is 5. This means that the point (5, 19) lies on the graph of the function F(x) = x² - 2x + 4. The process of simplifying expressions is a core skill in algebra and calculus. It allows us to manipulate mathematical expressions into simpler forms, making them easier to understand and work with. This skill is essential for solving equations, graphing functions, and performing other mathematical tasks. Therefore, mastering the art of simplifying expressions is crucial for success in mathematics.
The Answer
Therefore, F(5) = 19. This corresponds to option C in the given choices. We have successfully evaluated the function F(x) = x² - 2x + 4 at the point x = 5. This process involved substituting the value 5 for x in the function's expression and then simplifying the resulting expression using the order of operations. The final result, 19, represents the output of the function when the input is 5. It is important to verify the answer by reviewing the steps taken and ensuring that no arithmetic errors were made. In this case, we can double-check our calculations to confirm that 5 squared is indeed 25, 2 times 5 is 10, and 25 minus 10 plus 4 equals 19. This verification process helps to build confidence in the accuracy of the answer. The ability to evaluate functions accurately is a fundamental skill in mathematics, and it is used extensively in various applications. Whether we are graphing functions, solving equations, or modeling real-world phenomena, we often need to evaluate functions at specific points. Therefore, mastering this skill is crucial for success in mathematics and related fields. In addition to finding the value of a function at a specific point, we can also analyze the function's behavior over a range of values. This involves understanding how the output of the function changes as the input changes. This analysis can provide valuable insights into the function's properties, such as its maximum and minimum values, its intervals of increase and decrease, and its end behavior.
Conclusion
In this article, we have demonstrated how to find the value of a function at a specific point. By walking through the steps to find F(5) for the function F(x) = x² - 2x + 4, we have highlighted the importance of substitution and the order of operations. This is a fundamental skill in mathematics, applicable across various fields. The ability to evaluate functions accurately is essential for solving problems in algebra, calculus, and other areas of mathematics. It also plays a crucial role in various applications, such as modeling real-world phenomena and making predictions based on mathematical relationships. The process of evaluating a function involves several key steps: understanding the function's definition, substituting the input value for the variable, and simplifying the resulting expression using the order of operations. Each of these steps is important, and errors in any step can lead to an incorrect result. Therefore, it is crucial to pay close attention to detail and double-check calculations to ensure accuracy. In addition to the specific example we have worked through, the general principles and techniques discussed in this article can be applied to evaluate any function at any point. Whether the function is a simple linear function or a more complex trigonometric or exponential function, the process of substitution and simplification remains the same. Therefore, mastering these skills will provide a solid foundation for further studies in mathematics and related fields.
