How To Find The Domain Of A Rational Function F(x) = -3 / (x^2 - 2x - 35)

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In mathematics, determining the domain of a function is a fundamental task. The domain represents the set of all possible input values (often denoted as x) for which the function produces a valid output. When dealing with rational expressions, which are fractions where the numerator and denominator are polynomials, finding the domain requires special attention. This guide provides a comprehensive explanation of how to find the domain of a rational expression, using the example function f(x) = -3 / (x^2 - 2x - 35).

Understanding Rational Expressions and Domains

Rational expressions are mathematical expressions of the form P(x) / Q(x), where P(x) and Q(x) are polynomials. Polynomials are expressions consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. Examples of polynomials include x^2 - 2x - 35, 3x + 2, and 5. The key characteristic of a rational expression is that it involves division by a polynomial.

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For most polynomial functions, the domain is all real numbers, as there are no restrictions on the values that x can take. However, rational expressions introduce a critical restriction: the denominator cannot be equal to zero. Division by zero is undefined in mathematics, so any value of x that makes the denominator zero must be excluded from the domain.

Steps to Find the Domain of a Rational Expression

To find the domain of a rational expression, we follow a systematic approach:

  1. Identify the Denominator: The first step is to identify the denominator of the rational expression. This is the polynomial expression in the bottom part of the fraction. In our example, f(x) = -3 / (x^2 - 2x - 35), the denominator is x^2 - 2x - 35.
  2. Set the Denominator Equal to Zero: The next step is to set the denominator equal to zero. This equation will help us find the values of x that make the denominator zero, which we need to exclude from the domain. So, we have the equation x^2 - 2x - 35 = 0.
  3. Solve for x: Now, we need to solve the equation we obtained in the previous step. This will give us the values of x that make the denominator zero. The equation x^2 - 2x - 35 = 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 method. We look for two numbers that multiply to -35 and add to -2. These numbers are -7 and 5. So, we can factor the quadratic equation as (x - 7)(x + 5) = 0.
  4. Determine the Values to Exclude: Setting each factor equal to zero gives us the solutions: * x - 7 = 0 implies x = 7 * x + 5 = 0 implies x = -5 These values, x = 7 and x = -5, make the denominator zero and must be excluded from the domain.
  5. Express the Domain in Interval Notation: The final step is to express the domain in interval notation. Interval notation is a way of writing sets of numbers using intervals. Since we need to exclude x = 7 and x = -5 from the domain, the domain includes all real numbers except these two values. In interval notation, this is written as (-∞, -5) ∪ (-5, 7) ∪ (7, ∞). Here, the symbol ∪ represents the union of sets, and the parentheses indicate that the endpoints are not included in the interval. The intervals (-∞, -5), (-5, 7), and (7, ∞) represent all real numbers less than -5, between -5 and 7, and greater than 7, respectively. By combining these intervals, we include all real numbers except -5 and 7.

Detailed Solution for f(x) = -3 / (x^2 - 2x - 35)

Let's apply the steps outlined above to find the domain of the function f(x) = -3 / (x^2 - 2x - 35).

  1. Identify the Denominator: The denominator is x^2 - 2x - 35.
  2. Set the Denominator Equal to Zero: We set the denominator equal to zero: x^2 - 2x - 35 = 0.
  3. Solve for x: We factor the quadratic equation: (x - 7)(x + 5) = 0. This gives us two possible solutions for x: * x - 7 = 0 implies x = 7 * x + 5 = 0 implies x = -5
  4. Determine the Values to Exclude: The values x = 7 and x = -5 make the denominator zero, so we must exclude them from the domain.
  5. Express the Domain in Interval Notation: The domain is all real numbers except -5 and 7. In interval notation, this is written as (-∞, -5) ∪ (-5, 7) ∪ (7, ∞).

Thus, the domain of the rational expression f(x) = -3 / (x^2 - 2x - 35) is (-∞, -5) ∪ (-5, 7) ∪ (7, ∞).

Common Mistakes and How to Avoid Them

When finding the domain of rational expressions, several common mistakes can occur. Understanding these mistakes and how to avoid them can help you ensure accuracy.

  1. Forgetting to Exclude Values that Make the Denominator Zero: The most common mistake is forgetting to identify and exclude values that make the denominator zero. Always remember that division by zero is undefined, and any such values must be excluded from the domain. To avoid this, make it a habit to set the denominator equal to zero and solve for x.
  2. Incorrectly Solving the Equation: Another common mistake is incorrectly solving the equation obtained when setting the denominator equal to zero. This can happen due to errors in factoring, applying the quadratic formula, or other algebraic manipulations. Double-check your work and use alternative methods if possible to verify your solutions.
  3. Misinterpreting Interval Notation: Interval notation can be confusing for some students. Make sure you understand the meaning of parentheses and brackets. Parentheses indicate that the endpoint is not included, while brackets indicate that it is included. When expressing the domain of a rational expression, you will typically use parentheses to exclude the values that make the denominator zero.
  4. Not Factoring the Denominator Completely: Sometimes, the denominator may have more than two factors, especially if it is a higher-degree polynomial. Make sure to factor the denominator completely to identify all values that make it zero. This may involve using techniques such as polynomial long division or synthetic division.
  5. Ignoring the Numerator: While the numerator does not directly affect the domain of a rational expression, it is essential to consider it when analyzing other aspects of the function, such as its zeros and asymptotes. However, for domain determination, the focus remains solely on the denominator.

Examples and Practice Problems

To solidify your understanding, let's work through a few more examples and practice problems.

Example 1: Find the domain of g(x) = (x + 2) / (x - 3).

  1. Identify the Denominator: The denominator is x - 3.
  2. Set the Denominator Equal to Zero: x - 3 = 0
  3. Solve for x: x = 3
  4. Determine the Values to Exclude: We exclude x = 3.
  5. Express the Domain in Interval Notation: The domain is (-∞, 3) ∪ (3, ∞).

Example 2: Find the domain of h(x) = 5 / (x^2 - 4).

  1. Identify the Denominator: The denominator is x^2 - 4.
  2. Set the Denominator Equal to Zero: x^2 - 4 = 0
  3. Solve for x: Factor as (x - 2)(x + 2) = 0. This gives x = 2 and x = -2.
  4. Determine the Values to Exclude: We exclude x = 2 and x = -2.
  5. Express the Domain in Interval Notation: The domain is (-∞, -2) ∪ (-2, 2) ∪ (2, ∞).

Practice Problems:

  1. Find the domain of k(x) = 2x / (x + 1).
  2. Find the domain of m(x) = -1 / (x^2 - 9).
  3. Find the domain of n(x) = (x - 4) / (x^2 + 5x + 6).

By working through these examples and practice problems, you can enhance your skills in finding the domains of rational expressions.

Conclusion

Finding the domain of a rational expression is a crucial skill in algebra and calculus. By following the steps outlined in this guide, you can systematically determine the domain of any rational expression. Remember to identify the denominator, set it equal to zero, solve for x, and exclude those values from the domain. Expressing the domain in interval notation provides a clear and concise way to represent the set of all possible input values. With practice and careful attention to detail, you can confidently tackle domain-related problems in mathematics. Understanding these concepts thoroughly will serve as a solid foundation for more advanced topics in mathematics and related fields.