Solving (x+1)/(x^2-2x-3) = 1 A Step-by-Step Guide

Introduction to Rational Functions

In the realm of mathematics, rational functions play a crucial role in various fields, from calculus to engineering. Understanding how to solve equations involving rational functions is a fundamental skill. In this article, we will delve into solving a specific rational function equation: f(x) = (x+1)/(x^2-2x-3) = 1. We'll break down the steps, explain the underlying concepts, and ensure a clear, comprehensive understanding. This exploration is essential for anyone studying algebra, pre-calculus, or related disciplines, as it reinforces the principles of algebraic manipulation and equation-solving. Before we dive into the solution, let's first define what a rational function is and why solving such equations is important.

A rational function is essentially a function that can be expressed as the quotient of two polynomials. In simpler terms, it's a fraction where both the numerator and the denominator are polynomials. The given function, f(x) = (x+1)/(x^2-2x-3), perfectly fits this definition. The numerator, x+1, is a polynomial of degree 1 (a linear polynomial), and the denominator, x^2-2x-3, is a polynomial of degree 2 (a quadratic polynomial). Solving equations involving rational functions is crucial because it helps us find the values of x for which the function equals a specific value, in this case, 1. These values, often called solutions or roots, are significant in various applications, including finding the points where a graph intersects a horizontal line, determining equilibrium points in models, and solving optimization problems. The process involves algebraic manipulations to eliminate the fraction and transform the equation into a more manageable form, usually a polynomial equation. However, it's essential to remember to check for extraneous solutions, which are solutions obtained algebraically but do not satisfy the original equation due to division by zero. This careful approach ensures accurate and reliable results.

Solving rational functions also reinforces key algebraic concepts such as factoring, simplifying expressions, and solving polynomial equations. These skills are foundational for more advanced topics in mathematics and are widely applicable in various scientific and engineering disciplines. Therefore, mastering the techniques for solving rational function equations is not just about finding solutions to specific problems; it's about building a robust mathematical toolkit that can be applied across a wide range of contexts. This detailed exploration of the solution process will not only provide the answer to the given equation but also enhance your overall understanding of rational functions and their applications.

Step-by-Step Solution

To solve the rational function equation f(x) = (x+1)/(x^2-2x-3) = 1, we will follow a series of algebraic steps. Each step is designed to simplify the equation and isolate the variable x. The process involves eliminating the fraction, simplifying the resulting polynomial equation, and then solving for x. It's essential to pay close attention to each step to avoid common mistakes and ensure accuracy. The initial step is to eliminate the denominator, which will transform the rational equation into a standard polynomial equation. This transformation is a crucial step in making the equation solvable.

1. Eliminate the Denominator

The first step in solving the equation (x+1)/(x^2-2x-3) = 1 is to eliminate the denominator. We achieve this by multiplying both sides of the equation by the denominator, which is x^2-2x-3. This operation is valid as long as x^2-2x-3 is not equal to zero, as multiplying by zero would make the equation undefined. Multiplying both sides by x^2-2x-3 gives us: (x+1) = 1 * (x^2-2x-3). This step effectively removes the fraction, making the equation easier to manipulate. The resulting equation is a polynomial equation, which we can then simplify further. It's important to note that by multiplying both sides by an expression containing x, we introduce the possibility of extraneous solutions. These are solutions that satisfy the transformed equation but not the original equation. Therefore, we must check our solutions at the end of the process to ensure they are valid. This step is a fundamental technique in solving rational equations and is used extensively in various mathematical problems. The next step involves simplifying and rearranging the equation into a standard quadratic form, which we can then solve using various methods.

2. Simplify and Rearrange

After eliminating the denominator, we have the equation x+1 = x^2-2x-3. The next step is to rearrange this equation into a standard form, typically a quadratic equation in the form ax^2 + bx + c = 0. To do this, we subtract x and 1 from both sides of the equation. This gives us 0 = x^2 - 2x - 3 - x - 1, which simplifies to 0 = x^2 - 3x - 4. Now we have a quadratic equation in standard form, where a = 1, b = -3, and c = -4. This form is convenient because it allows us to use various methods to solve for x, such as factoring, completing the square, or using the quadratic formula. The process of rearranging the equation is crucial because it sets the stage for applying these solution methods. The standard form of a quadratic equation is well-studied, and there are established techniques for finding its roots. This step is a common practice in algebra and is used to solve various types of equations. The next step will involve solving this quadratic equation, which will give us the possible values of x that satisfy the equation. Factoring is often the first method to try when solving quadratic equations, as it can be a quick and straightforward approach if the equation factors nicely.

3. Solve the Quadratic Equation

Now that we have the quadratic equation x^2 - 3x - 4 = 0, we need to solve for x. One common method for solving quadratic equations is factoring. We look for two numbers that multiply to -4 (the constant term) and add up to -3 (the coefficient of the x term). These numbers are -4 and 1. Therefore, we can factor the quadratic equation as (x - 4)(x + 1) = 0. Setting each factor equal to zero gives us two possible solutions for x: x - 4 = 0 and x + 1 = 0. Solving these linear equations, we find x = 4 and x = -1. These are the potential solutions to the original rational equation. However, it's crucial to remember that we need to check these solutions to ensure they are not extraneous. Extraneous solutions can arise when we multiply both sides of an equation by an expression containing x, as we did in the first step. These solutions may satisfy the transformed equation but not the original equation due to division by zero. Therefore, the next step is to check these solutions in the original equation to ensure their validity. This step is a critical part of solving rational equations and cannot be overlooked.

4. Check for Extraneous Solutions

After finding the potential solutions x = 4 and x = -1, it is crucial to check for extraneous solutions. These are solutions that satisfy the transformed equation but not the original equation due to division by zero. The original equation is f(x) = (x+1)/(x^2-2x-3) = 1. We need to ensure that the denominator, x^2-2x-3, is not equal to zero for our solutions. First, let's factor the denominator: x^2-2x-3 = (x-3)(x+1). This shows that the denominator is zero when x = 3 or x = -1. Now, we can check our potential solutions. For x = 4, the denominator is (4-3)(4+1) = 1 * 5 = 5, which is not zero. Plugging x = 4 into the original equation, we get (4+1)/(4^2-2*4-3) = 5/(16-8-3) = 5/5 = 1, which satisfies the equation. Therefore, x = 4 is a valid solution. For x = -1, the denominator is (-1-3)(-1+1) = -4 * 0 = 0. Since the denominator is zero, x = -1 is an extraneous solution and must be discarded. This is a critical step in solving rational equations because extraneous solutions can lead to incorrect answers. The process of checking for extraneous solutions ensures that we only accept valid solutions. Therefore, the only valid solution to the equation is x = 4. This careful analysis highlights the importance of checking solutions in the original equation when dealing with rational functions.

Final Answer

After meticulously solving the rational function equation f(x) = (x+1)/(x^2-2x-3) = 1 and checking for extraneous solutions, we arrive at the final answer. The steps included eliminating the denominator, simplifying and rearranging the equation into a quadratic form, solving the quadratic equation, and verifying the solutions. This process is a comprehensive approach to solving rational equations and ensures that we obtain the correct result. The importance of checking for extraneous solutions cannot be overstated, as it prevents us from including invalid solutions in our final answer. The detailed explanation of each step provides a clear understanding of the solution process and the underlying mathematical principles.

The only valid solution to the equation is x = 4. The potential solution x = -1 was found to be extraneous because it made the denominator of the original equation equal to zero, which is undefined. Therefore, we discard x = -1 as a solution. The final answer is a single value, x = 4, which satisfies the original equation without causing any undefined operations. This solution can be verified by substituting x = 4 back into the original equation, which confirms that f(4) = 1. The process of solving rational equations involves a combination of algebraic manipulation and careful verification, and this example demonstrates the importance of both. The final answer, x = 4, is the only value that makes the equation true and is not an extraneous solution.

Therefore, the solution to the equation f(x) = (x+1)/(x^2-2x-3) = 1 is 4.