Domain Of The Rational Function F(x) = (x+1)/(x^2 - 2x - 3)

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In the realm of mathematics, understanding the domain of a function is paramount. The domain essentially defines the set of all possible input values (often x-values) for which the function produces a valid output. When dealing with rational functions, which are functions expressed as a ratio of two polynomials, determining the domain requires careful consideration of the denominator. This article delves into the process of finding the domain of the rational function f(x) = (x+1)/(x² - 2x - 3), providing a step-by-step explanation and highlighting the key concepts involved. By the end of this discussion, you will have a solid understanding of how to identify the domain of rational functions and the significance of excluding values that lead to undefined expressions.

Before we dive into the specifics of our given function, let's establish a clear understanding of rational functions and the concept of a domain. A rational function is any function that can be written in the form f(x) = P(x) / Q(x), where P(x) and Q(x) are polynomials. Polynomials are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. For instance, x + 1 and x² - 2x - 3 are both polynomials.

The domain of a function, as mentioned earlier, is the set of all permissible input values (x-values) for which the function yields a real number output. For most polynomial functions, the domain is all real numbers since we can substitute any real number into the polynomial and obtain a real number result. However, rational functions introduce a crucial restriction. Because division by zero is undefined in mathematics, any value of x that makes the denominator Q(x) equal to zero must be excluded from the domain. This is the core principle we'll use to determine the domain of f(x) = (x+1)/(x² - 2x - 3).

To find the domain of the given rational function, f(x) = (x+1)/(x² - 2x - 3), our primary task is to identify any values of x that make the denominator, x² - 2x - 3, equal to zero. These values must be excluded from the domain because they would result in division by zero, rendering the function undefined. The process involves the following steps:

  1. Set the denominator equal to zero: x² - 2x - 3 = 0

  2. Solve the quadratic equation: The equation x² - 2x - 3 = 0 is a quadratic equation, which can be solved by factoring, completing the square, or using the quadratic formula. In this case, factoring is the most straightforward approach. We need to find two numbers that multiply to -3 and add up to -2. These numbers are -3 and 1. Thus, we can factor the quadratic as follows: (x - 3)(x + 1) = 0

  3. Determine the roots: For the product of two factors to be zero, at least one of the factors must be zero. Therefore, we have two possible solutions:

    • x - 3 = 0 => x = 3
    • x + 1 = 0 => x = -1

  4. Exclude the roots from the domain: The values x = 3 and x = -1 make the denominator zero, so they must be excluded from the domain of the function. This is because substituting these values into the function would lead to division by zero, which is undefined.

  5. Express the domain: The domain of f(x) consists of all real numbers except x = 3 and x = -1. We can express this in several ways:

    • Set Notation: {x ∈ ℝ | x ≠ 3, x ≠ -1}
    • Interval Notation: (-∞, -1) ∪ (-1, 3) ∪ (3, ∞)

While factoring is an efficient method for solving quadratic equations when possible, it's not always the easiest or most applicable approach. Other methods include:

  • Quadratic Formula: The quadratic formula is a general solution that can be used to solve any quadratic equation of the form ax² + bx + c = 0. The formula is: x = (-b ± √(b² - 4ac)) / 2a For our equation x² - 2x - 3 = 0, a = 1, b = -2, and c = -3. Substituting these values into the quadratic formula, we get: x = (2 ± √((-2)² - 4 * 1 * -3)) / (2 * 1) x = (2 ± √(4 + 12)) / 2 x = (2 ± √16) / 2 x = (2 ± 4) / 2 This gives us two solutions: x = (2 + 4) / 2 = 3 and x = (2 - 4) / 2 = -1, which are the same values we obtained by factoring.

  • Completing the Square: Completing the square is another method for solving quadratic equations. It involves manipulating the equation to form a perfect square trinomial on one side. The steps are more involved than factoring, but it can be a useful technique, especially for equations that are difficult to factor. While we won't go through the full steps here, it's worth noting as another viable method.

As we mentioned earlier, the domain can be expressed in different notations. Understanding these notations is crucial for clear communication in mathematics.

  • Set Notation: This notation uses curly braces {} to define a set. The vertical bar | is read as "such that." So, {x ∈ ℝ | x ≠ 3, x ≠ -1} is read as "the set of all x belonging to the real numbers such that x is not equal to 3 and x is not equal to -1."

  • Interval Notation: This notation uses intervals to represent sets of numbers. Parentheses () indicate that the endpoint is not included, while brackets [] indicate that the endpoint is included. The symbol ∞ represents infinity. The union symbol ∪ is used to combine intervals. So, (-∞, -1) ∪ (-1, 3) ∪ (3, ∞) represents all real numbers less than -1, all real numbers between -1 and 3, and all real numbers greater than 3.

  • Graphical Representation: The domain can also be represented graphically on a number line. We would draw a number line and mark the points -1 and 3 with open circles (to indicate that they are not included) and then shade the regions to the left of -1, between -1 and 3, and to the right of 3. This visual representation provides a clear understanding of the values that are included in the domain.

Determining the domain of a function is not merely a mathematical exercise; it has significant implications in various applications. The domain defines the meaningful input values for a function within a given context. For instance, if a function models the population of a species over time, the domain might be restricted to non-negative values since time cannot be negative. Similarly, if a function models the area of a rectangle with a fixed perimeter, the domain would be limited by the physical constraints of the rectangle's dimensions.

In the case of rational functions, understanding the domain is crucial for avoiding undefined expressions and interpreting the behavior of the function near the excluded values. The values excluded from the domain often correspond to vertical asymptotes on the graph of the function. A vertical asymptote is a vertical line that the graph approaches but never touches. In our example, the graph of f(x) = (x+1)/(x² - 2x - 3) has vertical asymptotes at x = 3 and x = -1. This knowledge helps us understand how the function behaves as x approaches these values.

It's important to note that sometimes, factors in the numerator and denominator of a rational function can cancel out. This simplification can affect the appearance of the function's graph. In our example, f(x) = (x+1)/(x² - 2x - 3) = (x+1)/((x-3)(x+1)). The factor (x+1) appears in both the numerator and the denominator. If we were to simplify the function by canceling out the (x+1) terms, we would get a new function, g(x) = 1/(x-3). However, it's crucial to remember that the original function f(x) is still undefined at x = -1, even though it doesn't appear as a vertical asymptote in the simplified function g(x). Instead, there is a "hole" or a removable discontinuity at x = -1 on the graph of f(x).

A hole occurs when a factor cancels out, but the value that makes that factor zero is still not in the domain of the original function. This means that there is a point missing from the graph at that x-value. In our example, there is a hole at x = -1. To find the y-coordinate of the hole, we can substitute x = -1 into the simplified function g(x) = 1/(x-3), which gives us g(-1) = 1/(-1-3) = -1/4. So, the hole is located at the point (-1, -1/4).

In conclusion, determining the domain of a rational function is a fundamental skill in mathematics. For the rational function f(x) = (x+1)/(x² - 2x - 3), we found that the domain consists of all real numbers except x = 3 and x = -1. This was achieved by setting the denominator equal to zero, solving the resulting quadratic equation, and excluding the solutions from the domain. Understanding the domain not only helps us avoid undefined expressions but also provides valuable insights into the behavior and graph of the function, including vertical asymptotes and holes. Remember to always consider the restrictions imposed by the denominator when working with rational functions. By mastering this concept, you will be well-equipped to analyze and interpret a wide range of mathematical functions and their applications.

Answer: All real numbers except x = 3 and x = -1.