Evaluating F(3) For The Function F(x) = 1/(x-2) + 1
In this article, we will delve into the intricacies of evaluating a given function. Specifically, we will focus on the function f(x) = 1/(x-2) + 1. Our main objective is to determine the value of f(3). This involves substituting the value 3 for x in the function and simplifying the expression. This exercise is fundamental in understanding function notation and the basic operations involved in evaluating functions, crucial skills in mathematics and various fields that utilize mathematical modeling.
Understanding the Function Notation
Before we dive into the calculation, it's essential to grasp the concept of function notation. The notation f(x) represents a function named f that takes x as an input. The expression on the right-hand side of the equation defines the rule or operation that the function performs on the input. In our case, the function f(x) takes an input x, subtracts 2 from it, calculates the reciprocal of the result, and then adds 1. Understanding this notation is key to correctly interpreting and evaluating functions. The ability to work with function notation is a cornerstone of mathematical literacy, enabling us to express complex relationships and operations in a concise and standardized way. This understanding extends beyond simple substitution; it forms the basis for more advanced concepts like composite functions, inverse functions, and the application of functions in calculus and other areas of mathematics.
Step-by-Step Evaluation of f(3)
Now, let's proceed with evaluating f(3). This involves a straightforward process of substituting x with 3 in the function's expression:
- Substitute x with 3: Replace every instance of x in the function's equation with the value 3. This gives us f(3) = 1/(3-2) + 1.
- Simplify the denominator: Perform the subtraction in the denominator: 3 - 2 = 1. The expression now becomes f(3) = 1/1 + 1.
- Calculate the reciprocal: Divide 1 by 1, which results in 1. The expression is now f(3) = 1 + 1.
- Perform the addition: Add 1 and 1 to get the final result. f(3) = 2.
Therefore, the value of the function f(x) = 1/(x-2) + 1 when x is 3 is 2. This step-by-step approach ensures clarity and accuracy in evaluating functions, particularly when dealing with more complex expressions. By breaking down the process into smaller, manageable steps, we minimize the risk of errors and gain a deeper understanding of how functions operate.
Importance of Order of Operations
In evaluating functions, it's crucial to adhere to the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order dictates the sequence in which mathematical operations should be performed to arrive at the correct answer. In our example, we first simplified the expression within the parentheses (3 - 2), then performed the division (1/1), and finally the addition (1 + 1). Failing to follow the order of operations can lead to incorrect results. For instance, if we were to add 1 to the fraction before dividing, we would arrive at a different answer. The order of operations is a fundamental principle in mathematics that ensures consistency and accuracy in calculations across all levels of complexity.
Domain and Range Considerations
While we successfully evaluated f(3), it's important to consider the domain of the function. The domain of a function is the set of all possible input values for which the function is defined. In our case, the function f(x) = 1/(x-2) + 1 is undefined when the denominator (x-2) is equal to zero, as division by zero is not allowed. Therefore, x cannot be 2. The domain of this function is all real numbers except 2, which can be expressed in interval notation as (-∞, 2) ∪ (2, ∞). Understanding the domain is crucial because it tells us which values we can safely input into the function without encountering undefined results.
Furthermore, the range of a function is the set of all possible output values. For f(x) = 1/(x-2) + 1, the range is all real numbers except 1. This is because the term 1/(x-2) can take on any real value except 0 (since a fraction can only equal 0 if its numerator is 0, and in this case, the numerator is 1). Therefore, f(x) can take on any value except 1 (since 0 + 1 = 1). The range provides insight into the possible output values of the function, which is essential for applications such as graphing and solving equations.
Applications of Function Evaluation
Function evaluation is not just a mathematical exercise; it has practical applications in various fields. For example, in physics, functions are used to model the motion of objects, and evaluating these functions at specific times allows us to determine the object's position or velocity. In economics, functions can represent supply and demand curves, and evaluating these functions helps economists predict market equilibrium. In computer science, functions are the building blocks of programs, and evaluating them is essential for executing code and producing results. The ability to evaluate functions is a fundamental skill that underpins many real-world applications.
Conclusion: Mastering Function Evaluation
In conclusion, evaluating the function f(x) = 1/(x-2) + 1 at x = 3 demonstrates the core principles of function notation and evaluation. By carefully substituting the value, simplifying the expression using the order of operations, and considering the function's domain, we arrived at the result f(3) = 2. This process highlights the importance of understanding function notation, adhering to the order of operations, and recognizing the limitations imposed by the function's domain. Mastering these concepts is crucial for success in mathematics and various fields that rely on mathematical modeling. The skills acquired through function evaluation provide a foundation for more advanced topics, such as calculus, differential equations, and mathematical analysis. The ability to work with functions is a valuable asset in any field that involves quantitative analysis and problem-solving.
