Finding The Domain Of F(x) = (x-2)/(x^2 + 8x - 9)

Jul 14, 2025 by ADMIN 50 views

Find the Domain of the Function f(x) = (x-2)/(x^2 + 8x - 9)

The domain of a function is the set of all possible input values (x-values) for which the function is defined. In simpler terms, it's all the x-values you can plug into the function and get a real number as an output. When we deal with rational functions, which are functions expressed as a ratio of two polynomials, we need to be particularly careful about the denominator. A rational function is undefined when its denominator is equal to zero, because division by zero is not allowed in mathematics. This concept is crucial when finding the domain of a rational function, as it helps us identify the x-values that must be excluded. Therefore, to find the domain of the given function, we must identify the values of x that make the denominator zero and exclude them from the set of all real numbers.

In the given function, f(x) = (x-2) / (x^2 + 8x - 9), the denominator is the quadratic expression x^2 + 8x - 9. To determine the domain, we need to find the values of x that make this expression equal to zero. These values will be the ones we must exclude from the domain. The process of finding these values involves setting the denominator equal to zero and solving for x. This typically involves factoring the quadratic expression or using the quadratic formula if factoring is not straightforward. By identifying the roots of the quadratic equation, we can pinpoint the x-values that lead to division by zero and consequently define the valid domain of the function. In the following sections, we will detail the steps involved in solving this quadratic equation and determining the domain of the function.

Rational functions, as the name suggests, involve ratios or fractions where the numerator and denominator are polynomials. When working with these functions, it's crucial to focus on the denominator. The primary issue that arises with rational functions is the possibility of the denominator becoming zero. This is a critical concern because division by zero is undefined in mathematics. Consequently, any x-value that makes the denominator zero must be excluded from the domain of the function. The domain of a rational function, therefore, consists of all real numbers except those that cause the denominator to equal zero.

To find the domain, the first step is to identify the denominator of the rational function. In our case, the function is f(x) = (x-2) / (x^2 + 8x - 9), so the denominator is the quadratic expression x^2 + 8x - 9. The next step is to set this denominator equal to zero and solve for x. This process will reveal the x-values that make the denominator zero, which are the values that are not allowed in the domain. Solving the equation x^2 + 8x - 9 = 0 is a crucial step in determining the domain of the function. The solutions to this equation will be the x-values that must be excluded. This can be achieved through various methods such as factoring, completing the square, or using the quadratic formula. Each of these methods offers a way to find the roots of the quadratic equation, which directly correspond to the x-values that are not part of the domain.

To determine the values of x that make the denominator zero, we set the quadratic expression x^2 + 8x - 9 equal to zero: x^2 + 8x - 9 = 0. This equation can be solved by factoring, which involves expressing the quadratic as a product of two binomials. We look for two numbers that multiply to -9 and add to 8. These numbers are 9 and -1. Thus, we can factor the quadratic as follows: (x + 9)(x - 1) = 0. The factored form of the equation allows us to easily identify the values of x that make the expression equal to zero. We set each factor equal to zero and solve for x:

x + 9 = 0 leads to x = -9
x - 1 = 0 leads to x = 1

These two values, x = -9 and x = 1, are the solutions to the equation x^2 + 8x - 9 = 0. This means that when x is -9 or 1, the denominator of the function f(x) becomes zero, making the function undefined at these points. Therefore, these values must be excluded from the domain of the function. Identifying these values is a crucial step in defining the valid inputs for the function. By excluding these values, we ensure that the function operates within the realm of real numbers, avoiding the undefined operation of division by zero. The next step involves expressing the domain in a suitable notation that clearly indicates the excluded values.

Having found the values x = -9 and x = 1 that make the denominator zero, we now know that these values must be excluded from the domain of the function f(x). The domain is the set of all real numbers except for these two values. To express this mathematically, we can use set notation. The domain can be written as the set of all x such that x is not equal to -9 and x is not equal to 1. In set-builder notation, this is written as:

{ x | x ≠ -9 and x ≠ 1 }

This notation clearly conveys that the domain includes all real numbers except -9 and 1. Alternatively, we can express the domain using interval notation. This involves representing the domain as a union of intervals. Since the domain includes all real numbers except -9 and 1, we can write it as the union of three intervals:

(-∞, -9) ∪ (-9, 1) ∪ (1, ∞)

This notation indicates that the domain includes all numbers less than -9, all numbers between -9 and 1, and all numbers greater than 1. Both set-builder notation and interval notation are valid ways to express the domain, and the choice between them often depends on the context or personal preference. The key is to accurately represent the set of all valid inputs for the function, ensuring that the excluded values are clearly indicated. In the final answer, we will typically choose the set-builder notation as it directly corresponds to the given options.

In conclusion, to find the domain of the function f(x) = (x-2) / (x^2 + 8x - 9), we identified the denominator, set it equal to zero, and solved for x. This yielded the values x = -9 and x = 1, which make the denominator zero and must be excluded from the domain. The domain of the function is therefore all real numbers except -9 and 1. Expressing this in set-builder notation, the domain is:

{ x | x ≠ -9 and x ≠ 1 }

This matches option B from the given choices. Therefore, the correct answer is:

B. { x | x ≠ -9 and x ≠ 1 }

Understanding the domain of a function is crucial in mathematics, especially when dealing with rational functions. This process involves identifying potential issues, such as division by zero, and excluding the corresponding values from the set of possible inputs. By following a systematic approach, we can accurately determine the domain and ensure that the function is well-defined for all valid inputs. This ensures that the function operates correctly and produces meaningful outputs. The domain is a fundamental concept in function analysis and is essential for a comprehensive understanding of mathematical functions.

While the main focus when finding the domain of a rational function is on the denominator, there are other types of functions where additional considerations might be necessary. For instance, when dealing with square root functions, the expression inside the square root must be non-negative (greater than or equal to zero) to yield real number outputs. Similarly, for logarithmic functions, the argument of the logarithm must be strictly positive (greater than zero). These restrictions arise from the nature of these mathematical operations and the constraints they impose on the input values. Therefore, when determining the domain of a function, it is essential to consider the specific type of function and any inherent limitations on its inputs.

For example, consider the function g(x) = √(x - 3). The domain of this function is all x such that x - 3 ≥ 0, which means x ≥ 3. Similarly, for the function h(x) = ln(x + 2), the domain is all x such that x + 2 > 0, which means x > -2. These additional considerations highlight the importance of understanding the properties of different types of functions when determining their domains. In each case, the goal is to identify the set of all possible input values that will produce valid, real number outputs. This comprehensive approach ensures that the domain is accurately determined, providing a complete picture of the function's behavior and limitations.