Domain Of (c*d)(x) With Given Functions: Explained!

igma-plugin-ds-dev-mode:on

Understanding the Domain of Combined Functions

When dealing with functions in mathematics, especially when combining them through operations like multiplication, it's crucial to understand the concept of the domain. The domain of a function refers to the set of all possible input values (often x-values) for which the function produces a valid output. In simpler terms, it's the range of 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. When we combine functions, the domain of the resulting function is influenced by the domains of the individual functions involved.

To find the domain of the combined function (cd)(x), we need to consider the domains of both c(x) and d(x). We must identify any values of x that would make either function undefined and exclude those values from the domain of the combined function. This involves examining each function separately and then finding the intersection of their domains. It’s like ensuring that whatever value we input into the combined function doesn’t break any of the individual functions that make it up. Therefore, let's dive into the specifics of the given functions, c(x) and d(x), and see how their individual domains influence the domain of their product, (cd)(x).

In the following sections, we will break down the process step by step, ensuring a clear understanding of how to determine the domain of combined functions.

Analyzing the Function c(x) = 5/(x-2)

Let's start by analyzing the function c(x) = 5/(x-2). This function is a rational function, which means it's a fraction where the numerator and the denominator are both polynomials. The key consideration when dealing with rational functions is the denominator. We need to identify any values of x that would make the denominator equal to zero, as division by zero is undefined in mathematics. The domain of a rational function consists of all real numbers except those that make the denominator zero.

In our case, the denominator is (x-2). To find the values of x that make the denominator zero, we set (x-2) equal to zero and solve for x:

x - 2 = 0 x = 2

This tells us that when x is equal to 2, the denominator becomes zero, and the function c(x) is undefined. Therefore, we must exclude x = 2 from the domain of c(x). The domain of c(x) can be expressed in several ways. We can say that the domain is all real numbers except x = 2. Alternatively, we can use interval notation to represent the domain as (-∞, 2) ∪ (2, ∞). This notation indicates that the domain includes all real numbers less than 2 and all real numbers greater than 2, but not 2 itself.

Understanding the domain of c(x) is a critical step in finding the domain of (cd)(x)* because any value that is not in the domain of c(x) cannot be in the domain of (cd)(x)*. Next, we will analyze the function d(x) to determine its domain.

Analyzing the Function d(x) = x + 3

Now, let’s consider the function d(x) = x + 3. This function is a linear function, which is a type of polynomial function. Linear functions are among the simplest types of functions, and they have a straightforward domain. Unlike rational functions or functions involving square roots, linear functions do not have any restrictions on their input values. This means that we can plug in any real number for x, and the function will produce a valid output.

The domain of d(x) = x + 3 includes all real numbers. There are no values of x that would cause the function to be undefined. We can express this in several ways: we can say that the domain is all real numbers, or we can use interval notation and write the domain as (-∞, ∞). This notation indicates that the domain includes all real numbers from negative infinity to positive infinity.

Since the domain of d(x) is all real numbers, it does not impose any restrictions on the domain of the combined function (cd)(x)*. However, we still need to consider the restrictions imposed by the domain of c(x), which we analyzed in the previous section. Understanding the domain of each individual function is essential for accurately determining the domain of their combination.

In the next section, we will combine our findings about the domains of c(x) and d(x) to determine the domain of (cd)(x)*.

Determining the Domain of (c*d)(x)

Now that we have analyzed the domains of c(x) and d(x) individually, we can determine the domain of the combined function (cd)(x). The function (cd)(x) represents the product of the functions c(x) and d(x), which means (cd)(x) = c(x) * d(x). To find the domain of (cd)(x), we need to consider the restrictions imposed by both c(x) and d(x).

Recall that the domain of c(x) = 5/(x-2) is all real numbers except x = 2. This is because x = 2 makes the denominator of c(x) equal to zero, resulting in an undefined expression. The domain of d(x) = x + 3, on the other hand, is all real numbers. There are no restrictions on the input values for d(x).

The domain of the combined function (cd)(x)* is the intersection of the domains of c(x) and d(x). In other words, a value of x can only be in the domain of (cd)(x)* if it is in the domain of both c(x) and d(x). Since the domain of d(x) is all real numbers, it does not impose any additional restrictions. The only restriction comes from c(x), which excludes x = 2.

Therefore, the domain of (cd)(x)* is all real numbers except x = 2. This means that we can plug in any real number for x into (cd)(x), except for 2, as it would make the original function c(x) undefined. In interval notation, the domain of (cd)(x) is (-∞, 2) ∪ (2, ∞).

Conclusion

In conclusion, to find the domain of the combined function (cd)(x), where c(x) = 5/(x-2) and d(x) = x + 3, we analyzed the domains of c(x) and d(x) individually. We found that the domain of c(x) is all real numbers except x = 2, and the domain of d(x) is all real numbers. The domain of (cd)(x) is the intersection of these domains, which is all real numbers except x = 2.

Understanding how to determine the domains of functions, especially when they are combined, is a fundamental concept in mathematics. It ensures that we are working with valid inputs and outputs and helps us avoid mathematical errors. By identifying and excluding values that would make a function undefined, we can accurately define the set of all possible input values for a function.

For further exploration of domain and range in functions, you can visit Khan Academy's page on domain and range. This resource provides additional explanations, examples, and practice problems to help you solidify your understanding of these concepts. Understanding function domains is a crucial aspect of mathematical analysis, and continued learning in this area will enhance your problem-solving skills.