Domain Of G(x) = (x^2 - 16) / (x - 4) In Interval Notation

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Introduction

In mathematics, 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 the collection of all 'x' values that you can plug into the function without causing any mathematical errors, such as division by zero or taking the square root of a negative number. Understanding the domain is crucial because it helps us define the boundaries within which the function operates meaningfully. This article aims to provide a comprehensive explanation of how to determine the domain of the function g(x) = (x^2 - 16) / (x - 4) and express it in interval notation. This involves identifying any values of x that would make the function undefined, such as values that lead to division by zero. The process includes factoring, simplifying the function, and then carefully considering the values of x that must be excluded from the domain. By the end of this discussion, you should have a clear understanding of how to find and represent the domain of rational functions, a fundamental concept in algebra and calculus.

Understanding the Function g(x)

To begin, let's take a closer look at the given function: g(x) = (x^2 - 16) / (x - 4). This function is a rational function, which means it is a ratio of two polynomials. The numerator is x^2 - 16, and the denominator is x - 4. When dealing with rational functions, one of the primary concerns is identifying values of x that make the denominator equal to zero. Division by zero is undefined in mathematics, and any such values must be excluded from the domain of the function. Therefore, to find the domain, we need to determine the values of x for which the denominator, x - 4, is not equal to zero. This is a crucial first step in understanding the function's behavior and its limitations. Factoring the numerator can sometimes simplify the function and provide additional insights into potential restrictions on the domain. For instance, in this case, the numerator x^2 - 16 can be factored into (x - 4)(x + 4), which reveals a common factor with the denominator. However, even though we can simplify the function algebraically, we must still consider the original form when determining the domain. The initial form of the function dictates the values that are permissible inputs.

Identifying Potential Restrictions

The main restriction we need to consider for the function g(x) = (x^2 - 16) / (x - 4) is when the denominator is equal to zero. As mentioned earlier, division by zero is undefined, so any value of x that makes the denominator zero must be excluded from the domain. To find these values, we set the denominator equal to zero and solve for x: x - 4 = 0. Adding 4 to both sides of the equation, we get x = 4. This means that when x is 4, the denominator of the function is zero, making the function undefined at this point. Therefore, x = 4 cannot be included in the domain of g(x). This single value creates a "hole" in the domain, which we must account for when expressing the domain in interval notation. It's also worth noting that while we can simplify the function by canceling out common factors (which we will discuss in the next section), the restriction imposed by the original denominator still applies. Even if the simplified form of the function appears to be defined at x = 4, the original function is not, and this must be reflected in the domain. Recognizing and addressing these restrictions is a fundamental aspect of working with rational functions.

Simplifying the Function

Before we proceed further, let's simplify the function g(x) = (x^2 - 16) / (x - 4). We can factor the numerator, x^2 - 16, as a difference of squares: x^2 - 16 = (x - 4)(x + 4). Now, the function becomes g(x) = [(x - 4)(x + 4)] / (x - 4). We can see that there is a common factor of (x - 4) in both the numerator and the denominator. Canceling this common factor, we get the simplified form: g(x) = x + 4. It's important to note that while this simplification is mathematically valid, it doesn't change the original restriction on the domain. The original function was undefined at x = 4, and this restriction still holds true even for the simplified form. The simplified function, g(x) = x + 4, is a linear function, which is defined for all real numbers. However, because of the original function's form, we must exclude x = 4 from the domain. This is a critical concept in dealing with rational functions: simplification can make the function easier to analyze and graph, but it doesn't eliminate the restrictions imposed by the original function's denominator. Understanding this distinction is vital for correctly determining and expressing the domain.

Expressing the Domain in Interval Notation

Now that we have identified the restriction on the domain of g(x) = (x^2 - 16) / (x - 4), we can express the domain in interval notation. We know that x cannot be equal to 4, but it can be any other real number. Interval notation is a way of writing sets of numbers using intervals, which are ranges of values. To express the domain of g(x), we need to represent all real numbers except for 4. This can be done using two intervals: one that includes all numbers less than 4, and another that includes all numbers greater than 4. The interval for numbers less than 4 is written as (-∞, 4), where the parenthesis indicates that 4 is not included in the interval. Similarly, the interval for numbers greater than 4 is written as (4, ∞), again with a parenthesis to exclude 4. To combine these two intervals into a single expression for the domain, we use the union symbol, ∪. The union of two sets includes all elements from both sets. Therefore, the domain of g(x) in interval notation is (-∞, 4) ∪ (4, ∞). This notation concisely represents all real numbers except 4, which is precisely the domain of the function g(x). Using interval notation is a standard way to express domains and ranges in mathematics, providing a clear and unambiguous representation of the possible values.

Conclusion

In summary, determining the domain of a function is a fundamental concept in mathematics, particularly when dealing with rational functions. For the function g(x) = (x^2 - 16) / (x - 4), we identified that the denominator, x - 4, cannot be equal to zero. This restriction led us to exclude x = 4 from the domain. Even though the function can be simplified to g(x) = x + 4, the original restriction still applies. Expressing the domain in interval notation, we represented all real numbers except 4 as (-∞, 4) ∪ (4, ∞). This notation provides a clear and concise way to communicate the set of all possible input values for the function. Understanding how to find and express the domain of functions is crucial for further studies in algebra, calculus, and other areas of mathematics. It ensures that we are working with valid input values and can accurately interpret the behavior of functions. The process involves identifying potential restrictions, simplifying the function if possible, and then using interval notation to represent the domain. By mastering these steps, you can confidently determine the domain of a wide range of functions.