Domain Of Rational Function F(x) = 2x(x-1) / (3x^2-23x-36) Explained
When working with rational functions, determining the domain is a crucial first step. The domain of a function represents all possible input values (x-values) for which the function is defined. For rational functions, which are essentially fractions with polynomials in the numerator and denominator, the main concern is avoiding division by zero. This guide provides a step-by-step explanation of how to find the domain of the rational function F(x) = 2x(x-1) / (3x^2 - 23x - 36). We'll delve into the process of identifying the values that make the denominator zero and excluding them from the domain, thus ensuring that the function operates within the bounds of mathematical validity. Understanding the domain is paramount for graphing, analyzing, and applying rational functions in various mathematical and real-world contexts. It helps to establish the boundaries within which the function behaves predictably and avoid any undefined results. In essence, finding the domain is about understanding the function's limitations and ensuring that it is used appropriately.
Understanding Rational Functions and Domains
To effectively find the domain of a rational function, it's essential to first grasp the fundamental concepts involved. A rational function is any 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. For example, F(x) = 2x(x-1) / (3x^2 - 23x - 36) is a rational function because both 2x(x-1) and (3x^2 - 23x - 36) are polynomials. Polynomials themselves are algebraic expressions that consist of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
The domain of a function, on the other hand, is the set of all possible input values (x-values) for which the function produces a valid output. For most polynomial functions, the domain is all real numbers because there are no restrictions on the values you can plug in. However, rational functions introduce a critical exception: division by zero. Since division by zero is undefined in mathematics, any value of x that makes the denominator of a rational function equal to zero must be excluded from the domain. This is the core principle behind finding the domain of a rational function – identifying and excluding the values that lead to an undefined operation. Understanding this principle is crucial for avoiding mathematical errors and for accurately interpreting the behavior of the function. The process involves setting the denominator equal to zero, solving for x, and then stating the domain as all real numbers except those solutions.
Step 1: Identify the Denominator
The first crucial step in finding the domain of the given rational function, F(x) = 2x(x-1) / (3x^2 - 23x - 36), is to pinpoint the denominator. The denominator is the expression located in the bottom part of the fraction, which in this case is 3x^2 - 23x - 36. Identifying the denominator is paramount because it is the key to determining the values of x that would make the function undefined. Remember, a rational function is undefined when the denominator equals zero, as division by zero is not a permissible operation in mathematics. Once the denominator is correctly identified, the next step involves setting it equal to zero and solving for x. This process will reveal the x-values that must be excluded from the domain of the function. Accurate identification of the denominator is essential for the subsequent steps, as any error here will lead to an incorrect determination of the domain. Therefore, careful attention must be paid to this initial step to ensure the final result is valid.
Step 2: Set the Denominator Equal to Zero
Once we've identified the denominator of our rational function, which is 3x^2 - 23x - 36, the next pivotal step in finding the domain is to set this expression equal to zero. This action translates the problem into solving the equation 3x^2 - 23x - 36 = 0. The reason for this step is rooted in the fundamental principle that a rational function is undefined when its denominator is zero. By setting the denominator to zero, we are essentially seeking the x-values that would cause this undefined situation. These x-values are precisely the ones that must be excluded from the function's domain. The equation 3x^2 - 23x - 36 = 0 is a quadratic equation, and solving it will reveal the specific values of x that make the denominator zero. This is a critical step in the process, as the solutions obtained here directly influence the final determination of the domain. Therefore, careful attention and accurate algebraic manipulation are essential to ensure the correct identification of these crucial x-values.
Step 3: Solve the Quadratic Equation
Having set the denominator equal to zero, we now face the task of solving the quadratic equation 3x^2 - 23x - 36 = 0. There are several methods to tackle this, including factoring, using the quadratic formula, or completing the square. In this case, factoring is a viable approach. We need to find two numbers that multiply to (3)(-36) = -108 and add up to -23. These numbers are -27 and 4. So, we can rewrite the middle term of the quadratic equation as follows: 3x^2 - 27x + 4x - 36 = 0. Next, we factor by grouping: 3x(x - 9) + 4(x - 9) = 0. This gives us (3x + 4)(x - 9) = 0. Now, we set each factor equal to zero and solve for x: 3x + 4 = 0, which yields x = -4/3, and x - 9 = 0, which yields x = 9. These two values, x = -4/3 and x = 9, are the solutions to the quadratic equation. They are the critical values that make the denominator of the rational function equal to zero. Therefore, they must be excluded from the domain to avoid division by zero. The accurate solution of the quadratic equation is a pivotal step in determining the domain of the function.
Step 4: Determine the Domain
After solving the quadratic equation and finding the values that make the denominator zero, we are now ready to determine the domain of the rational function F(x) = 2x(x-1) / (3x^2 - 23x - 36). We found that the solutions to 3x^2 - 23x - 36 = 0 are x = -4/3 and x = 9. This means that when x is either -4/3 or 9, the denominator of the rational function becomes zero, resulting in an undefined expression. To define the domain, we must exclude these values from the set of all real numbers. The domain can be expressed in several ways. In set notation, it is {x | x ∈ ℝ, x ≠ -4/3, x ≠ 9}, which reads as “the set of all x such that x is a real number and x is not equal to -4/3 or 9.” Alternatively, we can use interval notation to represent the domain as (-∞, -4/3) ∪ (-4/3, 9) ∪ (9, ∞). This notation indicates that the domain includes all real numbers less than -4/3, all real numbers between -4/3 and 9, and all real numbers greater than 9. Both notations accurately convey the domain of the function, which is all real numbers except for x = -4/3 and x = 9. Understanding and correctly expressing the domain is essential for interpreting the function's behavior and its applicability in various contexts.
Expressing the Domain
Having identified the values that must be excluded, the final step involves formally expressing the domain of the rational function. There are two common ways to represent the domain: set notation and interval notation. Set notation provides a concise way to define the set of all permissible x-values. For our function, F(x) = 2x(x-1) / (3x^2 - 23x - 36), where we found that x cannot be -4/3 or 9, the set notation representation of the domain is: {x | x ∈ ℝ, x ≠ -4/3, x ≠ 9}. This notation is read as