The Domain Of $f(x)$ Is The Set Of All Real Values Except 7, And The Domain Of $g(x)$ Is The Set Of All Real Values Except -3.Which Of The Following Describes The Domain Of $(g \circ F)(x)$?A. All Real Values Except

Introduction

In mathematics, the concept of domain is crucial when dealing with functions. The domain of a function is the set of all possible input values for which the function is defined. When working with composite functions, it's essential to understand how the domains of individual functions interact. In this article, we'll explore the domain of composite functions, specifically the function $(g \circ f)(x)$, where the domains of $f(x)$ and $g(x)$ are given.

Understanding the Domain of Individual Functions

Before diving into the domain of composite functions, let's review the domains of individual functions $f(x)$ and $g(x)$.

Domain of $f(x)$

The domain of $f(x)$ is the set of all real values except 7. This means that $f(x)$ is defined for all real numbers except $x = 7$. In other words, the domain of $f(x)$ is $\mathbb{R} \setminus \{7\}$.

Domain of $g(x)$

Similarly, the domain of $g(x)$ is the set of all real values except -3. This implies that $g(x)$ is defined for all real numbers except $x = -3$. In other words, the domain of $g(x)$ is $\mathbb{R} \setminus \{-3\}$.

The Domain of Composite Functions

Now that we've established the domains of individual functions, let's consider the composite function $(g \circ f)(x)$. The composite function is defined as $(g \circ f)(x) = g(f(x))$. To find the domain of $(g \circ f)(x)$, we need to consider the restrictions imposed by both $f(x)$ and $g(x)$.

Restrictions Imposed by $f(x)$

Since the domain of $f(x)$ is $\mathbb{R} \setminus \{7\}$, we know that $f(x)$ is not defined at $x = 7$. This means that any input value $x$ that results in $f(x) = 7$ will also be excluded from the domain of $(g \circ f)(x)$.

Restrictions Imposed by $g(x)$

Similarly, since the domain of $g(x)$ is $\mathbb{R} \setminus \{-3\}$, we know that $g(x)$ is not defined at $x = -3$. This means that any input value $x$ that results in $f(x) = -3$ will also be excluded from the domain of $(g \circ f)(x)$.

Combining the Restrictions

To find the domain of $(g \circ f)(x)$, we need to combine the restrictions imposed by both $f(x)$ and $g(x)$. This means that we need to exclude all input values $x$ that result in either $f(x) = 7$ or $f(x) = -3$.

Excluding Input Values

Let's consider the two cases:

• If $f(x) = 7$, then $x$ is excluded from the domain of $(g \circ f)(x)$.
• If $f(x) = -3$, then $x$ is also excluded from the domain of $(g \circ f)(x)$.

Combining the Exclusions

To find the domain of $(g \circ f)(x)$, we need to combine these two exclusions. This means that we need to exclude all input values $x$ that result in either $f(x) = 7$ or $f(x) = -3$.

The Final Domain

After combining the restrictions imposed by both $f(x)$ and $g(x)$, we can conclude that the domain of $(g \circ f)(x)$ is the set of all real values except 7 and -3. In other words, the domain of $(g \circ f)(x)$ is $\mathbb{R} \setminus \{7, -3\}$.

Conclusion

In conclusion, the domain of composite functions is determined by combining the restrictions imposed by individual functions. By understanding the domains of individual functions and combining the restrictions, we can determine the domain of composite functions. In this article, we've explored the domain of the composite function $(g \circ f)(x)$, where the domains of $f(x)$ and $g(x)$ are given. We've shown that the domain of $(g \circ f)(x)$ is the set of all real values except 7 and -3.

Final Answer

$$

Q1: What is the domain of a composite function?

A1: 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 values that can be plugged into the function without resulting in an undefined or invalid output.

Q2: How do I find the domain of a composite function?

A2: To find the domain of a composite function, you need to combine the restrictions imposed by individual functions. This means that you need to exclude all input values that result in either a function being undefined or an invalid output.

Q3: What are some common restrictions that can affect the domain of a composite function?

A3: Some common restrictions that can affect the domain of a composite function include:

• Division by zero
• Taking the square root of a negative number
• Evaluating a function at a value that is not in its domain
• Evaluating a function at a value that results in an undefined or invalid output

Q4: How do I determine the domain of a composite function with multiple restrictions?

A4: To determine the domain of a composite function with multiple restrictions, you need to combine the restrictions imposed by individual functions. This means that you need to exclude all input values that result in either a function being undefined or an invalid output.

Q5: Can the domain of a composite function be a single value?

A5: Yes, the domain of a composite function can be a single value. For example, if the composite function is defined only at a single point, then the domain of the composite function is that single point.

Q6: Can the domain of a composite function be an interval?

A6: Yes, the domain of a composite function can be an interval. For example, if the composite function is defined on a specific interval, then the domain of the composite function is that interval.

Q7: How do I graph the domain of a composite function?

A7: To graph the domain of a composite function, you need to identify the restrictions imposed by individual functions and exclude the corresponding input values. This can be done by plotting the restrictions on a number line or a graph.

Q8: Can the domain of a composite function be expressed using interval notation?

A8: Yes, the domain of a composite function can be expressed using interval notation. For example, if the domain of the composite function is all real numbers except 7 and -3, then the domain can be expressed as $\mathbb{R} \setminus \{7, -3\}$.

Q9: How do I determine the domain of a composite function with a variable in the denominator?

A9: To determine the domain of a composite function with a variable in the denominator, you need to exclude all input values that result in the denominator being equal to zero.

Q10: Can the domain of a composite function be expressed using set notation?

A10: Yes, the domain of a composite function can be expressed using set notation. For example, if the domain of the composite function is all real numbers except 7 and -3, then the domain can be expressed as $\{x \in \mathbb{R} \mid x \neq 7, x \neq -3\}$.

Conclusion

In conclusion, the domain of a composite function is determined by combining the restrictions imposed by individual functions. By understanding the domains of individual functions and combining the restrictions, we can determine the domain of composite functions. In this article, we've explored some frequently asked questions about the domain of composite functions and provided answers to help you better understand this concept.

Final Answer

The final answer is: $\boxed{\mathbb{R} \setminus \{7, -3\}}$