If $c(x)=\frac{5}{x-2}$ And $d(x)=x+3$, What Is The Domain Of $(c \cdot D)(x$\]?A. All Real Values Of $x$ B. All Real Values Of $x$ Except $x=2$ C. All Real Values Of $x$ Except

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When dealing with composite functions, it's essential to understand the domain of each individual function involved. In this case, we have two functions: c(x)=5xβˆ’2c(x)=\frac{5}{x-2} and d(x)=x+3d(x)=x+3. We're asked to find the domain of the composite function (cβ‹…d)(x)(c \cdot d)(x).

The Domain of a Function

The domain of a function is the set of all possible input values for which the function is defined. In other words, it's the set of all possible values of xx that can be plugged into the function without causing any issues.

The Domain of c(x)c(x)

Let's start by analyzing the domain of c(x)=5xβˆ’2c(x)=\frac{5}{x-2}. This function has a denominator of xβˆ’2x-2, which means that we cannot have x=2x=2 as an input value, as it would result in division by zero. Therefore, the domain of c(x)c(x) is all real values of xx except x=2x=2.

The Domain of d(x)d(x)

Next, let's consider the domain of d(x)=x+3d(x)=x+3. This function is a simple linear function, and it's defined for all real values of xx. In other words, there are no restrictions on the input values for d(x)d(x).

The Domain of (cβ‹…d)(x)(c \cdot d)(x)

Now that we've analyzed the domains of c(x)c(x) and d(x)d(x), we can determine the domain of the composite function (cβ‹…d)(x)(c \cdot d)(x). Since the composite function is the product of c(x)c(x) and d(x)d(x), we need to consider the restrictions on the input values imposed by both functions.

Restrictions on the Input Values

As we mentioned earlier, the function c(x)c(x) is not defined at x=2x=2, as it would result in division by zero. However, the function d(x)d(x) is defined for all real values of xx. Therefore, the only restriction on the input values for the composite function (cβ‹…d)(x)(c \cdot d)(x) is that xx cannot be equal to 22.

Conclusion

In conclusion, the domain of the composite function (cβ‹…d)(x)(c \cdot d)(x) is all real values of xx except x=2x=2. This is because the function c(x)c(x) is not defined at x=2x=2, and the function d(x)d(x) is defined for all real values of xx.

Answer

The correct answer is B. All real values of xx except x=2x=2.

Additional Examples

To further illustrate the concept of composite functions and their domains, let's consider a few additional examples.

Example 1

Suppose we have two functions: f(x)=1xβˆ’1f(x)=\frac{1}{x-1} and g(x)=x2g(x)=x^2. What is the domain of the composite function (fβ‹…g)(x)(f \cdot g)(x)?

Solution

The domain of f(x)f(x) is all real values of xx except x=1x=1. The domain of g(x)g(x) is all real values of xx. Therefore, the domain of the composite function (fβ‹…g)(x)(f \cdot g)(x) is all real values of xx except x=1x=1.

Example 2

Suppose we have two functions: h(x)=1x+2h(x)=\frac{1}{x+2} and j(x)=xj(x)=\sqrt{x}. What is the domain of the composite function (hβ‹…j)(x)(h \cdot j)(x)?

Solution

The domain of h(x)h(x) is all real values of xx except x=βˆ’2x=-2. The domain of j(x)j(x) is all non-negative real values of xx. Therefore, the domain of the composite function (hβ‹…j)(x)(h \cdot j)(x) is all non-negative real values of xx except x=βˆ’2x=-2.

Example 3

Suppose we have two functions: k(x)=1xβˆ’3k(x)=\frac{1}{x-3} and l(x)=x3l(x)=x^3. What is the domain of the composite function (kβ‹…l)(x)(k \cdot l)(x)?

Solution

The domain of k(x)k(x) is all real values of xx except x=3x=3. The domain of l(x)l(x) is all real values of xx. Therefore, the domain of the composite function (kβ‹…l)(x)(k \cdot l)(x) is all real values of xx except x=3x=3.

Conclusion

In our previous article, we discussed the concept of composite functions and their domains. We analyzed the domains of individual functions and determined the domain of the composite function. In this article, we'll answer some frequently asked questions about the domain of composite functions.

Q: What is the domain of a composite function?

A: The domain of a composite function is the set of all possible input values for which the function is defined. In other words, it's the set of all possible values of xx that can be plugged into the function without causing any issues.

Q: How do I determine the domain of a composite function?

A: To determine the domain of a composite function, you need to analyze the domains of the individual functions involved. You should identify any restrictions on the input values imposed by each function and determine the intersection of these restrictions.

Q: What are some common restrictions on the input values of composite functions?

A: Some common restrictions on the input values of composite functions include:

  • Division by zero: This occurs when the denominator of a fraction is equal to zero.
  • Square root of a negative number: This occurs when the input value is negative and the function involves a square root.
  • Logarithm of a non-positive number: This occurs when the input value is non-positive and the function involves a logarithm.

Q: How do I handle restrictions on the input values of composite functions?

A: To handle restrictions on the input values of composite functions, you should:

  • Identify the restrictions imposed by each individual function.
  • Determine the intersection of these restrictions.
  • Express the domain of the composite function using interval notation.

Q: What is interval notation?

A: Interval notation is a way of expressing the domain of a function using intervals. It's a shorthand way of writing the domain of a function and is commonly used in mathematics.

Q: How do I express the domain of a composite function using interval notation?

A: To express the domain of a composite function using interval notation, you should:

  • Identify the restrictions imposed by each individual function.
  • Determine the intersection of these restrictions.
  • Express the domain of the composite function using interval notation.

Q: What are some examples of composite functions and their domains?

A: Here are some examples of composite functions and their domains:

  • (fβ‹…g)(x)=1xβˆ’1β‹…x2(f \cdot g)(x) = \frac{1}{x-1} \cdot x^2, where f(x)=1xβˆ’1f(x) = \frac{1}{x-1} and g(x)=x2g(x) = x^2. The domain of this composite function is all real values of xx except x=1x=1.
  • (hβ‹…j)(x)=1x+2β‹…x(h \cdot j)(x) = \frac{1}{x+2} \cdot \sqrt{x}, where h(x)=1x+2h(x) = \frac{1}{x+2} and j(x)=xj(x) = \sqrt{x}. The domain of this composite function is all non-negative real values of xx except x=βˆ’2x=-2.
  • (kβ‹…l)(x)=1xβˆ’3β‹…x3(k \cdot l)(x) = \frac{1}{x-3} \cdot x^3, where k(x)=1xβˆ’3k(x) = \frac{1}{x-3} and l(x)=x3l(x) = x^3. The domain of this composite function is all real values of xx except x=3x=3.

Conclusion

In conclusion, the domain of a composite function is determined by the restrictions on the input values imposed by each individual function involved. By analyzing the domains of the individual functions and determining the intersection of these restrictions, we can determine the domain of the composite function.