Evaluating The Function F(x) = (x-1)/(x^2+2) At X = 1/3

Introduction

In this article, we will delve into the process of evaluating a function at a specific point. Functions are a fundamental concept in mathematics, representing a relationship between an input and an output. When we evaluate a function, we substitute a given value for the input variable and perform the operations defined by the function to determine the corresponding output. This process is crucial in various mathematical and real-world applications, allowing us to understand the behavior of functions and make predictions based on their properties.

Our focus will be on the function f(x) = (x-1)/(x^2+2). This is a rational function, which means it is defined as the ratio of two polynomials. Rational functions are common in mathematics and physics, modeling phenomena such as the motion of projectiles, the decay of radioactive substances, and the flow of fluids. To gain a deeper understanding of this particular function, we will evaluate it at x = 1/3. This exercise will not only demonstrate the mechanics of function evaluation but also provide insights into the function's behavior at a specific point. By substituting 1/3 for x in the function's expression, we will simplify the resulting expression to obtain the function's value at that point. This process involves basic arithmetic operations, such as fraction manipulation and order of operations, which are essential skills in mathematics.

Evaluating functions is a cornerstone of mathematical analysis. It enables us to determine the output of a function for a given input, which is essential for graphing functions, solving equations, and understanding the function's behavior. The process involves substituting the input value into the function's expression and simplifying to obtain the output value. In the case of f(x) = (x-1)/(x^2+2), evaluating it at x = 1/3 will provide us with a specific numerical value that represents the function's output when the input is 1/3. This value can then be used for various purposes, such as plotting the function on a graph or comparing it with other function values. Moreover, understanding how to evaluate functions is a prerequisite for more advanced topics in calculus and mathematical modeling.

Step-by-Step Evaluation of f(1/3)

To evaluate the function f(x) = (x-1)/(x^2+2) at x = 1/3, we need to substitute 1/3 for every instance of x in the function's expression. This is the fundamental principle of function evaluation: replace the variable with the given value and then simplify the expression. The substitution will transform the function's expression into a numerical expression that we can then simplify using the order of operations. This step is critical because it sets the stage for the rest of the evaluation process. A correct substitution ensures that we are working with the correct numerical values and relationships, which is essential for obtaining an accurate result. Careful attention to detail is crucial at this stage to avoid any errors in the substitution process. Once the substitution is complete, we will have a numerical expression that we can then simplify using arithmetic operations.

After substituting 1/3 for x, we get f(1/3) = ((1/3) - 1) / ((1/3)^2 + 2). This expression now contains fractions and exponents, so we need to follow the order of operations (PEMDAS/BODMAS) to simplify it correctly. The order of operations dictates the sequence in which mathematical operations should be performed: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Adhering to this order is essential for obtaining the correct result. First, we will simplify the terms within the parentheses in both the numerator and the denominator. This involves subtracting 1 from 1/3 in the numerator and squaring 1/3 in the denominator. These initial simplifications will make the expression easier to manage and pave the way for further calculations.

Next, we simplify the numerator and the denominator separately. In the numerator, we have (1/3) - 1. To subtract these numbers, we need to express 1 as a fraction with a denominator of 3, which gives us 3/3. Then, we can subtract the fractions: (1/3) - (3/3) = -2/3. In the denominator, we have (1/3)^2 + 2. First, we need to square 1/3, which gives us (1/9). Then, we add 2 to 1/9. Again, we need to express 2 as a fraction with a denominator of 9, which gives us 18/9. So, (1/9) + (18/9) = 19/9. Now, our expression looks like f(1/3) = (-2/3) / (19/9). This step involves basic arithmetic operations with fractions, which are fundamental skills in mathematics. Accurate simplification of the numerator and denominator is crucial for obtaining the correct final answer.

Finally, we divide the simplified numerator by the simplified denominator. Dividing by a fraction is the same as multiplying by its reciprocal. So, we multiply (-2/3) by the reciprocal of (19/9), which is (9/19). This gives us (-2/3) * (9/19) = (-2 * 9) / (3 * 19) = -18/57. We can further simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3. This gives us (-18/3) / (57/3) = -6/19. Therefore, f(1/3) = -6/19. This final simplification step ensures that we have the answer in its simplest form, which is often preferred in mathematical contexts. The result, -6/19, represents the value of the function f(x) when x is equal to 1/3. This value can be used for various purposes, such as graphing the function or comparing it with other function values.

Detailed Calculation

Let's revisit the evaluation process with a detailed breakdown of each step. We start with the function f(x) = (x-1)/(x^2+2) and the input value x = 1/3. Our goal is to find the value of f(1/3), which means substituting 1/3 for every instance of x in the function's expression. This substitution is the first and most crucial step in the evaluation process. A correct substitution ensures that we are working with the correct numerical values and relationships, which is essential for obtaining an accurate result. Careful attention to detail is crucial at this stage to avoid any errors in the substitution process. Once the substitution is complete, we will have a numerical expression that we can then simplify using arithmetic operations.

Substituting x = 1/3 into the function, we get:

f(1/3) = ((1/3) - 1) / ((1/3)^2 + 2)

Now, we need to simplify this expression using the order of operations (PEMDAS/BODMAS). First, we simplify the terms within the parentheses in both the numerator and the denominator. This involves subtracting 1 from 1/3 in the numerator and squaring 1/3 in the denominator. These initial simplifications will make the expression easier to manage and pave the way for further calculations. The order of operations dictates the sequence in which mathematical operations should be performed: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Adhering to this order is essential for obtaining the correct result.

Simplifying the numerator:

(1/3) - 1 = (1/3) - (3/3) = -2/3

To subtract 1 from 1/3, we express 1 as a fraction with a denominator of 3, which gives us 3/3. Then, we subtract the fractions: (1/3) - (3/3) = -2/3. This step involves basic arithmetic operations with fractions, which are fundamental skills in mathematics. Accurate simplification of the numerator and denominator is crucial for obtaining the correct final answer.

Simplifying the denominator:

(1/3)^2 + 2 = (1/9) + 2 = (1/9) + (18/9) = 19/9

First, we square 1/3, which gives us (1/9). Then, we add 2 to 1/9. Again, we need to express 2 as a fraction with a denominator of 9, which gives us 18/9. So, (1/9) + (18/9) = 19/9. This step also involves basic arithmetic operations with fractions, which are fundamental skills in mathematics. Accurate simplification of the numerator and denominator is crucial for obtaining the correct final answer.

Now, we have:

f(1/3) = (-2/3) / (19/9)

Dividing by a fraction is the same as multiplying by its reciprocal. So, we multiply (-2/3) by the reciprocal of (19/9), which is (9/19). This gives us:

f(1/3) = (-2/3) * (9/19) = (-2 * 9) / (3 * 19) = -18/57

Finally, we simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3:

f(1/3) = (-18/3) / (57/3) = -6/19

Therefore, f(1/3) = -6/19. This final simplification step ensures that we have the answer in its simplest form, which is often preferred in mathematical contexts. The result, -6/19, represents the value of the function f(x) when x is equal to 1/3. This value can be used for various purposes, such as graphing the function or comparing it with other function values.

Conclusion

In conclusion, we have successfully evaluated the function f(x) = (x-1)/(x^2+2) at x = 1/3. The process involved substituting 1/3 for x in the function's expression and then simplifying the resulting numerical expression using the order of operations. This process is a fundamental skill in mathematics, as it allows us to determine the output of a function for a given input. The steps we followed included simplifying the numerator and denominator separately, and then dividing the simplified numerator by the simplified denominator. This involved basic arithmetic operations with fractions, such as subtraction, squaring, addition, and division. Careful attention to detail and adherence to the order of operations were crucial for obtaining the correct result.

The result of our evaluation is f(1/3) = -6/19. This value represents the output of the function f(x) when the input is 1/3. This specific value can be used for various purposes, such as plotting the function on a graph, comparing it with other function values, or using it in further calculations. Understanding how to evaluate functions is a prerequisite for more advanced topics in calculus and mathematical modeling. It enables us to analyze the behavior of functions, solve equations, and make predictions based on mathematical models.

The ability to evaluate functions is essential for various applications in mathematics, science, and engineering. Functions are used to model relationships between quantities, and evaluating a function allows us to determine the value of one quantity given the value of another. For example, in physics, functions are used to model the motion of objects, and evaluating the function at a specific time gives us the object's position at that time. In economics, functions are used to model supply and demand, and evaluating the function at a specific price gives us the quantity supplied or demanded at that price. In computer science, functions are used to perform specific tasks, and evaluating the function with certain inputs produces the desired output. Therefore, mastering the skill of function evaluation is crucial for success in these fields.