If $f(x) = X + 7$ And $g(x) = \frac{1}{x - 13}$, What Is The Domain Of $(f \circ G)(x$\]?A. $\{x \mid X \neq 6\}$ B. $\{x \mid X \neq -6\}$ C. $\{x \mid X \neq -13\}$ D. $\{x \mid X \neq

When dealing with composite functions, it's essential to understand the concept of the domain. The domain of a function is the set of all possible input values for which the function is defined. In this article, we will explore the domain of the composite function $(f \circ g)(x)$, where $f(x) = x + 7$ and $g(x) = \frac{1}{x - 13}$.

What is a Composite Function?

A composite function is a function that is derived from two or more functions. In this case, we have two functions, $f(x)$ and $g(x)$, and we want to find the composite function $(f \circ g)(x)$. The composite function is defined as:

$$(f \circ g)(x) = f(g(x))$$

Finding the Composite Function

To find the composite function, we need to substitute $g(x)$ into $f(x)$.

$$(f \circ g)(x) = f(g(x))$$

$$= f\left(\frac{1}{x - 13}\right)$$

$$= \frac{1}{x - 13} + 7$$

Understanding the Domain of the Composite Function

Now that we have the composite function, we need to find its domain. The domain of the composite function is the set of all possible input values for which the function is defined. In this case, we need to consider the restrictions on the domain of both $f(x)$ and $g(x)$.

Restrictions on the Domain of $g(x)$

The function $g(x) = \frac{1}{x - 13}$ has a restriction on its domain. The denominator of the function cannot be equal to zero, as this would result in an undefined value.

$$x - 13 \neq 0$$

$$x \neq 13$$

Restrictions on the Domain of $f(x)$

The function $f(x) = x + 7$ has no restrictions on its domain. However, we need to consider the restrictions on the domain of the composite function.

Finding the Domain of the Composite Function

Now that we have considered the restrictions on the domain of both $f(x)$ and $g(x)$, we can find the domain of the composite function. The composite function is defined as:

$$(f \circ g)(x) = \frac{1}{x - 13} + 7$$

The domain of the composite function is the set of all possible input values for which the function is defined. In this case, we need to consider the restriction on the domain of $g(x)$.

$$x \neq 13$$

However, we also need to consider the restriction on the domain of $f(x)$. The function $f(x)$ is defined for all real numbers, but the composite function is not defined when the denominator of $g(x)$ is equal to zero.

$$x - 13 \neq 0$$

$$x \neq 13$$

Conclusion

In conclusion, the domain of the composite function $(f \circ g)(x)$ is the set of all real numbers except $x = 13$. This is because the function $g(x)$ is not defined when the denominator is equal to zero, and the composite function is not defined when the denominator of $g(x)$ is equal to zero.

Answer

The correct answer is:

A. $\{x \mid x \neq 13\}$

Final Thoughts

In this article, we explored the domain of the composite function $(f \circ g)(x)$, where $f(x) = x + 7$ and $g(x) = \frac{1}{x - 13}$. We found that the domain of the composite function is the set of all real numbers except $x = 13$. This is because the function $g(x)$ is not defined when the denominator is equal to zero, and the composite function is not defined when the denominator of $g(x)$ is equal to zero.

References

• [1] "Composite Functions" by Math Open Reference
• [2] "Domain of a Function" by Math Is Fun

Additional Resources

• [1] "Composite Functions" by Khan Academy
• [2] "Domain of a Function" by Purplemath
  Q&A: Understanding the Domain of Composite Functions =====================================================

In our previous article, we explored the domain of the composite function $(f \circ g)(x)$, where $f(x) = x + 7$ and $g(x) = \frac{1}{x - 13}$. We found that the domain of the composite function is the set of all real numbers except $x = 13$. In this article, we will 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 is the set of all values of $x$ for which the composite function is defined.

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

A: To find the domain of a composite function, you need to consider the restrictions on the domain of both functions. You need to find the values of $x$ for which the denominator of the inner function is not equal to zero, and the values of $x$ for which the outer function is defined.

Q: What is the difference between the domain of a function and the domain of a composite function?

A: The domain of a function is the set of all possible input values for which the function is defined. The domain of a composite function is the set of all possible input values for which the composite function is defined. In other words, the domain of a composite function is a subset of the domain of the outer function.

Q: How do I determine the domain of a composite function with a rational function as the inner function?

A: To determine the domain of a composite function with a rational function as the inner function, you need to find the values of $x$ for which the denominator of the inner function is not equal to zero. You also need to consider the values of $x$ for which the outer function is defined.

Q: What is the domain of the composite function $(f \circ g)(x)$, where $f(x) = \frac{1}{x - 2}$ and $g(x) = \frac{1}{x + 3}$?

A: To find the domain of the composite function $(f \circ g)(x)$, we need to consider the restrictions on the domain of both functions. The function $g(x)$ is not defined when the denominator is equal to zero, so we need to find the values of $x$ for which $x + 3 \neq 0$. This gives us $x \neq -3$. The function $f(x)$ is defined for all real numbers except $x = 2$, so we need to consider the values of $x$ for which $x \neq 2$. Therefore, the domain of the composite function $(f \circ g)(x)$ is the set of all real numbers except $x = -3$ and $x = 2$.

Q: What is the domain of the composite function $(f \circ g)(x)$, where $f(x) = \frac{1}{x - 1}$ and $g(x) = \frac{1}{x - 4}$?

A: To find the domain of the composite function $(f \circ g)(x)$, we need to consider the restrictions on the domain of both functions. The function $g(x)$ is not defined when the denominator is equal to zero, so we need to find the values of $x$ for which $x - 4 \neq 0$. This gives us $x \neq 4$. The function $f(x)$ is defined for all real numbers except $x = 1$, so we need to consider the values of $x$ for which $x \neq 1$. Therefore, the domain of the composite function $(f \circ g)(x)$ is the set of all real numbers except $x = 1$ and $x = 4$.

Conclusion

In conclusion, the domain of a composite function is the set of all possible input values for which the function is defined. To find the domain of a composite function, you need to consider the restrictions on the domain of both functions. You need to find the values of $x$ for which the denominator of the inner function is not equal to zero, and the values of $x$ for which the outer function is defined.

References

• [1] "Composite Functions" by Math Open Reference
• [2] "Domain of a Function" by Math Is Fun

Additional Resources

• [1] "Composite Functions" by Khan Academy
• [2] "Domain of a Function" by Purplemath