Evaluating F(a+2) For F(x) = (3+x)/(x-3) A Step-by-Step Guide

Introduction to Function Evaluation

In the realm of mathematics, functions serve as fundamental building blocks, mapping inputs to corresponding outputs. Understanding how to evaluate functions for specific inputs is a crucial skill in various mathematical disciplines. In this article, we will delve into the function f(x) = (3+x)/(x-3) and explore the process of evaluating it at x = a+2. This exploration will not only enhance our understanding of function evaluation but also provide insights into algebraic manipulation and simplification. When dealing with functions, it's important to recognize that they represent a relationship between two sets of elements. The function f(x) takes an input x and transforms it according to a specific rule, in this case, (3+x)/(x-3). Evaluating a function means substituting a given value for the input variable and then simplifying the resulting expression. This process is fundamental to solving equations, graphing functions, and understanding their behavior. The function f(x) = (3+x)/(x-3) is a rational function, which means it is a ratio of two polynomials. Rational functions have unique characteristics, such as potential vertical asymptotes where the denominator equals zero. These functions are essential in calculus, algebra, and various applications, including physics and engineering. Understanding how to evaluate and manipulate rational functions is a critical skill for any student pursuing further studies in these fields. The expression f(a+2) means we are replacing every instance of x in the function's formula with the expression a+2. This substitution is the first step in evaluating the function at the given input. After the substitution, we will simplify the resulting expression using algebraic techniques to arrive at the final answer. This process may involve combining like terms, factoring, and canceling common factors. The ability to perform these algebraic manipulations accurately is crucial for success in mathematics.

Step-by-Step Evaluation of f(a+2)

To determine the value of f(a+2), we begin by substituting (a+2) for x in the function f(x) = (3+x)/(x-3). This yields f(a+2) = (3 + (a+2))/((a+2) - 3). This initial substitution is a fundamental step in evaluating any function at a specific input. It's crucial to ensure that the substitution is performed correctly, replacing every instance of the input variable with the given value or expression. In this case, we are replacing x with the binomial (a+2), so it is essential to use parentheses to maintain the correct order of operations. After the substitution, we simplify the expression by combining like terms in both the numerator and the denominator. In the numerator, we have 3 + (a+2), which simplifies to a + 5. In the denominator, we have (a+2) - 3, which simplifies to a - 1. Therefore, f(a+2) becomes (a + 5)/(a - 1). This simplification process is a critical part of evaluating functions, as it allows us to express the result in its most concise form. It involves applying the rules of algebra, such as the commutative and associative properties of addition, to combine terms effectively. Simplifying the numerator and denominator separately makes the expression easier to manage and reduces the risk of errors. The simplified expression (a + 5)/(a - 1) represents the value of the function f(x) when x is equal to a+2. This is a rational expression, and its value depends on the value of a. It's important to note that the function is undefined when the denominator is equal to zero, which occurs when a = 1. This value is excluded from the domain of the function. The ability to simplify algebraic expressions like this is essential in mathematics, as it allows us to solve equations, graph functions, and make generalizations about their behavior. Practice in simplifying expressions will help you to develop your algebraic skills and improve your understanding of mathematical concepts.

Analyzing the Result and Identifying the Correct Option

Having evaluated f(a+2) to be (a+5)/(a-1), we now turn our attention to the provided options to identify the one that matches our result. The options are:

A. (3+f(a+2))/(f(a)-1) B. (5+a)/(a-1) C. (3+a)/(a-3) + 2

Comparing our result, (a+5)/(a-1), with the options, we can immediately see that option B, (5+a)/(a-1), is equivalent to our simplified expression. The order of terms in addition does not affect the result, so (a+5) is the same as (5+a). Therefore, option B is the correct answer. This step of comparing the result with the given options is crucial in multiple-choice questions. It ensures that you have not made any errors in your calculations and that you have arrived at the correct answer. Sometimes, the options may be presented in a different form than your result, so it may be necessary to perform additional algebraic manipulations to verify the equivalence. In this case, the simplified form of f(a+2) directly matches option B, making it easy to identify the correct answer. However, it is still important to carefully examine each option to avoid making careless mistakes. Option A, (3+f(a+2))/(f(a)-1), involves the function evaluated at both a+2 and a. To verify whether this option is equivalent, we would need to substitute the expressions for f(a+2) and f(a) and simplify. This would be a more time-consuming process, but it could be used as a check if there was any doubt about the direct match with option B. Option C, (3+a)/(a-3) + 2, is a different form altogether. To compare this with our result, we would need to combine the terms by finding a common denominator. This would also be a more involved process, and it is clear that option C is not equivalent to our simplified expression. By carefully comparing the result with the options, we have confidently identified option B as the correct answer. This demonstrates the importance of not only being able to evaluate functions but also being able to analyze and compare algebraic expressions.

Verification and Alternative Approaches

While we have confidently arrived at the answer f(a+2) = (a+5)/(a-1) and identified option B as the correct choice, it's always beneficial to explore verification methods and consider alternative approaches. This not only reinforces our understanding but also provides a safety net against potential errors. One way to verify our result is to substitute a specific numerical value for a into both the original function with (a+2) as the input and the simplified expression. For instance, let's choose a = 2. Substituting a = 2 into the original function, we get f(2+2) = f(4) = (3+4)/(4-3) = 7/1 = 7. Now, substituting a = 2 into our simplified expression, (a+5)/(a-1), we get (2+5)/(2-1) = 7/1 = 7. Since both results match, this provides strong evidence that our simplification is correct. This numerical verification method is a valuable tool for checking algebraic manipulations. It involves choosing a specific value for the variable and evaluating the expressions before and after simplification. If the results match, it increases our confidence in the correctness of our work. However, it's important to note that numerical verification does not guarantee correctness for all values of the variable. It's possible, though unlikely, that the expressions could match for a specific value but not be equivalent in general. Another approach to verify our result would be to work backward from the simplified expression (a+5)/(a-1) and try to manipulate it back into the original form with the substitution x = a+2. This would involve reversing the steps of simplification, which can be a useful way to check for errors. For example, we could try to rewrite (a+5)/(a-1) in terms of (a+2). This approach is often more challenging but can provide a deeper understanding of the algebraic manipulations involved. Considering alternative approaches to solving a problem can also be beneficial. In this case, we directly substituted (a+2) into the function and simplified. However, another approach might be to first simplify the function algebraically and then substitute (a+2). This could potentially lead to a more efficient solution, depending on the specific function and the substitution involved. By exploring verification methods and considering alternative approaches, we not only strengthen our understanding of the problem but also develop our problem-solving skills in general.

Conclusion: Mastering Function Evaluation

In this article, we have thoroughly explored the evaluation of the function f(x) = (3+x)/(x-3) at x = a+2. By systematically substituting (a+2) for x, simplifying the resulting expression, and comparing our result with the given options, we confidently identified the correct answer as (a+5)/(a-1), which corresponds to option B. This process has highlighted the importance of several key mathematical skills, including function evaluation, algebraic manipulation, and simplification of expressions. Mastering function evaluation is crucial for success in various mathematical disciplines. It is a fundamental skill that underpins many other concepts, such as solving equations, graphing functions, and understanding their behavior. The ability to accurately substitute values into functions and simplify the resulting expressions is essential for both theoretical and applied mathematics. Algebraic manipulation is another critical skill that we have utilized throughout this exploration. This involves applying the rules of algebra to transform expressions into equivalent forms. Skills such as combining like terms, factoring, and simplifying fractions are essential for simplifying expressions and solving equations. Practice in algebraic manipulation will help you to develop your fluency in mathematics and improve your problem-solving abilities. The simplification of expressions is often the final step in function evaluation and other mathematical problems. It involves reducing an expression to its simplest form, which makes it easier to understand and work with. Simplifying expressions requires a strong understanding of algebraic rules and techniques. By working through this example, we have reinforced our understanding of these key mathematical skills. We have also explored verification methods and considered alternative approaches, which further enhance our problem-solving abilities. Function evaluation is a fundamental skill in mathematics, and by mastering it, you will be well-prepared for more advanced topics. This article provides a comprehensive guide to evaluating functions and highlights the importance of accuracy, simplification, and verification. By practicing these skills, you can build a strong foundation in mathematics and excel in your studies.

Repair Input Keyword: If f(x) = (3+x)/(x-3), what is f(a+2)?

Title: Evaluating f(a+2) for f(x) = (3+x)/(x-3) A Step-by-Step Guide