Solving F(x) = -1 For F(x) = 1/(x-2) + 1 A Comprehensive Guide
Introduction
In this article, we will delve into the process of finding the value of x that satisfies the equation f(x) = -1, where the function f(x) is defined as f(x) = 1/(x-2) + 1. This problem falls under the domain of mathematics, specifically focusing on the analysis of rational functions. To effectively solve this, we will utilize a combination of algebraic manipulation and graphical interpretation. Understanding rational functions is crucial in various fields, including calculus, engineering, and physics, making this exploration not only academically valuable but also practically relevant.
Understanding the Function
To begin, let's deeply understand the function f(x) = 1/(x-2) + 1. This is a rational function, which means it is a function that can be expressed as the quotient of two polynomials. In this case, the numerator is 1, and the denominator is (x-2). The function also includes a vertical asymptote at x = 2, a point where the function is undefined because the denominator becomes zero. This asymptote significantly influences the behavior of the function around x = 2. As x approaches 2 from the left, f(x) approaches negative infinity, and as x approaches 2 from the right, f(x) approaches positive infinity. This characteristic is a hallmark of rational functions with vertical asymptotes.
Moreover, the function has a horizontal asymptote at y = 1. This can be observed by analyzing the behavior of the function as x approaches positive or negative infinity. As x becomes very large (either positively or negatively), the term 1/(x-2) approaches zero, and f(x) approaches 1. The horizontal asymptote provides insight into the function's long-term behavior, indicating the value that f(x) tends towards as x moves away from the origin. Understanding these asymptotes is critical for accurately graphing the function and predicting its values.
Graphical Interpretation
Visualizing the function through a graph is a powerful tool for solving the problem. By plotting the function f(x) = 1/(x-2) + 1, we can see its behavior and identify where it intersects the line y = -1. The graph will show the vertical asymptote at x = 2 and the horizontal asymptote at y = 1. The curve of the function will approach these asymptotes but never cross them. The graph also helps in understanding the transformations applied to the basic reciprocal function 1/x. The function f(x) is essentially a transformation of 1/x, shifted 2 units to the right and 1 unit upwards. This understanding aids in sketching the graph without relying on a calculator.
The intersection point of the function's graph and the line y = -1 will give us the solution to the equation f(x) = -1. This graphical method provides a visual confirmation of the algebraic solution, ensuring a comprehensive understanding of the problem. By observing the graph, we can estimate the solution before performing algebraic calculations, which can help in verifying the accuracy of the final answer. The graphical approach not only solves the problem at hand but also enhances the understanding of the function's characteristics and behavior.
Solving for f(x) = -1
Algebraic Approach
To find the value of x when f(x) = -1, we need to solve the equation:
1/(x-2) + 1 = -1
First, let's isolate the term with x by subtracting 1 from both sides of the equation:
1/(x-2) = -2
Next, we multiply both sides by (x-2) to eliminate the fraction. This step is valid as long as x ≠2 (because we know x = 2 is a vertical asymptote and thus not a solution):
1 = -2(x-2)
Now, distribute the -2 on the right side:
1 = -2x + 4
Add 2x to both sides and subtract 1 from both sides to isolate the x term:
2x = 3
Finally, divide both sides by 2 to solve for x:
x = 3/2
Therefore, the value of x that satisfies f(x) = -1 is x = 3/2.
Verification
To ensure the accuracy of our solution, we can substitute x = 3/2 back into the original equation:
f(3/2) = 1/((3/2)-2) + 1
Simplify the denominator:
f(3/2) = 1/(-1/2) + 1
Now, divide 1 by -1/2, which is equivalent to multiplying 1 by -2:
f(3/2) = -2 + 1
f(3/2) = -1
Since the result is -1, our solution x = 3/2 is correct. This verification step is crucial in mathematical problem-solving to avoid errors and ensure a solid understanding of the solution.
Conclusion
In this article, we successfully found the value of x for which f(x) = -1, given the function f(x) = 1/(x-2) + 1. We approached the problem by first understanding the characteristics of the function, including its vertical and horizontal asymptotes. We then used an algebraic method to solve the equation, carefully manipulating the equation to isolate x. The solution we found was x = 3/2.
To ensure the correctness of our solution, we verified it by substituting x = 3/2 back into the original equation, which confirmed that f(3/2) indeed equals -1. This comprehensive approach, combining algebraic manipulation with verification, is essential for solving mathematical problems accurately.
This problem highlights the importance of understanding rational functions and their graphical representations. The asymptotes play a crucial role in defining the function's behavior, and visualizing the graph aids in predicting and confirming solutions. By mastering these concepts, one can confidently tackle similar problems in various mathematical and real-world contexts.
