Domain Of The Product Of Functions C(x) And D(x)

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In mathematics, determining the domain of a function is a fundamental concept. The domain of a function is the set of all possible input values (often represented by 'x') for which the function produces a valid output. When dealing with combinations of functions, such as the product of two functions, understanding how individual domains interact becomes crucial. This article delves into finding the domain of the product of two functions, specifically focusing on the functions c(x) = 5/(x-2) and d(x) = x + 3. Let's explore the step-by-step process to determine the domain of their product, (cd)(x).

Understanding the Functions c(x) and d(x)

Before we can determine the domain of the product of the functions, let's first understand the individual functions themselves. We are given two functions:

  • c(x) = 5/(x-2)
  • d(x) = x + 3

The function c(x) is a rational function, which means it is a ratio of two polynomials. In this case, the numerator is the constant 5, and the denominator is the linear expression x - 2. Rational functions have a unique characteristic: they are undefined when the denominator is equal to zero. This is because division by zero is undefined in mathematics. Therefore, to find the domain of c(x), we need to identify any values of x that make the denominator zero. Setting x - 2 = 0, we find that x = 2 is the value that makes the denominator zero. This means that c(x) is defined for all real numbers except x = 2.

The function d(x) = x + 3 is a linear function. Linear functions are defined for all real numbers. There are no restrictions on the values of x that can be input into this function. No matter what value of x you choose, you will always get a valid output.

Determining the Domain of c(x)

As we established earlier, the domain of c(x) is all real numbers except for x = 2. This is because when x = 2, the denominator of c(x) becomes 2 - 2 = 0, and division by zero is undefined. We can express this domain in several ways:

  • Set Notation: {x | x ∈ ℝ, x ≠ 2}
  • Interval Notation: (-∞, 2) ∪ (2, ∞)

Both notations convey the same meaning: the domain includes all real numbers less than 2 and all real numbers greater than 2, but it does not include 2 itself.

Understanding the restrictions on the domain of c(x) is crucial because it will directly impact the domain of the product function (cd)(x).

Determining the Domain of d(x)

The function d(x) = x + 3 is a linear function, and linear functions are remarkably straightforward when it comes to their domains. Unlike rational functions or radical functions, there are no denominators that could become zero, and no square roots that demand non-negative arguments. In essence, you can input any real number into d(x), and it will produce a valid, real number output.

This means that the domain of d(x) encompasses all real numbers. We can express this mathematically in the following ways:

  • Set Notation: {x | x ∈ ℝ}
  • Interval Notation: (-∞, ∞)

Both notations clearly indicate that there are no restrictions on the values of x that can be used in the function d(x). Any real number will work.

While the domain of d(x) itself is unrestricted, it's important to keep in mind that when we combine d(x) with other functions, such as c(x) in this case, the domain of the resulting function might be affected by the restrictions of the other function. This is precisely what we'll explore when we consider the product (cd)(x).

Defining the Product Function (cd)(x)

Now that we understand the individual functions c(x) and d(x), let's define their product, (cd)(x). The product of two functions is simply a new function formed by multiplying the two original functions together. In this case:

(cd)(x) = c(x) * d(x)

Substituting the given expressions for c(x) and d(x), we get:

(cd)(x) = (5/(x-2)) * (x + 3)

We can simplify this expression by multiplying the numerators and keeping the same denominator:

(cd)(x) = (5(x + 3))/(x - 2)

(cd)(x) = (5x + 15)/(x - 2)

This simplified expression shows that (cd)(x) is also a rational function. Like c(x), it has a denominator that can potentially be zero, which will restrict the domain of the function.

Understanding the form of (cd)(x) as a rational function is key to determining its domain. We need to identify any values of x that would make the denominator equal to zero, as those values will be excluded from the domain.

Identifying Restrictions on the Domain of (cd)(x)

To find the domain of the product function (cd)(x), we need to consider the restrictions imposed by each individual function, c(x) and d(x). Remember that the domain of a combined function is the intersection of the domains of the individual functions.

We already know that:

  • The domain of c(x) is all real numbers except x = 2.
  • The domain of d(x) is all real numbers.

Since d(x) has no restrictions, the only restriction on the domain of (cd)(x) comes from c(x). The function c(x) is undefined when its denominator, x - 2, is equal to zero. Therefore, we must exclude x = 2 from the domain of (cd)(x).

Alternatively, we can look at the simplified expression for (cd)(x) = (5x + 15)/(x - 2). The denominator is x - 2, and setting it equal to zero gives us x - 2 = 0, which means x = 2. This confirms that x = 2 must be excluded from the domain.

Therefore, the domain of (cd)(x) is all real numbers except x = 2.

Expressing the Domain of (cd)(x)

Now that we've identified the restrictions on the domain of (cd)(x), we can express the domain using different notations. As we've established, the domain includes all real numbers except for x = 2. Here are the common ways to represent this:

  • Set Notation: {x | x ∈ ℝ, x ≠ 2}
    • This notation reads as "the set of all x such that x is a member of the real numbers and x is not equal to 2."
  • Interval Notation: (-∞, 2) ∪ (2, ∞)
    • This notation uses intervals to represent the range of values. (-∞, 2) represents all numbers less than 2, and (2, ∞) represents all numbers greater than 2. The union symbol (∪) combines these two intervals, indicating that the domain includes both ranges but excludes the value 2 itself.

Both set notation and interval notation are precise ways to communicate the domain of (cd)(x). Understanding these notations is crucial for effectively working with functions and their domains in mathematics.

Conclusion

In conclusion, the domain of the product function (cd)(x), where c(x) = 5/(x-2) and d(x) = x + 3, is all real numbers except x = 2. This restriction arises from the function c(x), which is undefined when x = 2 due to division by zero. The function d(x) has no domain restrictions, so it does not contribute to the restrictions on the domain of the product. We can express the domain of (cd)(x) using set notation as {x | x ∈ ℝ, x ≠ 2} or interval notation as (-∞, 2) ∪ (2, ∞). Understanding how to determine the domain of combined functions is essential for working with mathematical functions and ensuring valid results.

When finding the domain of combined functions, always remember to consider the domains of the individual functions and identify any restrictions that might arise from operations like division by zero or taking the square root of a negative number. By carefully analyzing the functions involved, you can accurately determine the domain of the resulting function.