Function $g$ Is Defined As $g(x) = 2f(x - 4) + 3$. What Is The Domain Of Function $g$?A. $\{x \mid -4 \ \textless \ X \ \textless \ \infty\}$ B. $\{x \mid 0 \ \textless \ X \ \textless \ \infty\}$

When dealing with composite functions, it's essential to understand the relationship between the two functions involved. In this case, we're given the function $g(x) = 2f(x - 4) + 3$, and we're asked to find the domain of function $g$. To do this, we need to consider the domain of the inner function $f(x - 4)$ and how it affects the domain of the outer function $g(x)$.

Understanding the Inner Function $f(x - 4)$

The inner function $f(x - 4)$ is a transformation of the function $f(x)$, where the input $x$ is shifted 4 units to the right. This means that the domain of $f(x - 4)$ is the same as the domain of $f(x)$, but shifted 4 units to the right.

Domain of $f(x)$

Let's assume that the domain of $f(x)$ is $\{x \mid a \ \textless \ x \ \textless \ b\}$. Then, the domain of $f(x - 4)$ is $\{x \mid a + 4 \ \textless \ x \ \textless \ b + 4\}$.

Domain of $g(x)$

Now, let's consider the function $g(x) = 2f(x - 4) + 3$. Since the domain of $f(x - 4)$ is $\{x \mid a + 4 \ \textless \ x \ \textless \ b + 4\}$, the domain of $g(x)$ is the same as the domain of $f(x - 4)$, but shifted 0 units to the right (since there is no horizontal shift in the function $g(x)$).

Finding the Domain of $g(x)$

To find the domain of $g(x)$, we need to find the values of $x$ for which the expression $2f(x - 4) + 3$ is defined. Since the expression $2f(x - 4) + 3$ is a linear combination of the function $f(x - 4)$ and a constant, the domain of $g(x)$ is the same as the domain of $f(x - 4)$.

Conclusion

Based on the above analysis, we can conclude that the domain of function $g$ is $\{x \mid a + 4 \ \textless \ x \ \textless \ b + 4\}$. However, we are given two options for the domain of function $g$: $\{x \mid -4 \ \textless \ x \ \textless \ \infty\}$ and $\{x \mid 0 \ \textless \ x \ \textless \ \infty\}$. To determine which option is correct, we need to consider the possible values of $a$ and $b$.

Analyzing the Options

Let's analyze the two options:

• Option A: $\{x \mid -4 \ \textless \ x \ \textless \ \infty\}$
• Option B: $\{x \mid 0 \ \textless \ x \ \textless \ \infty\}$

Option A: $\{x \mid -4 \ \textless \ x \ \textless \ \infty\}$

If the domain of function $g$ is $\{x \mid -4 \ \textless \ x \ \textless \ \infty\}$, then we can conclude that $a + 4 = -4$ and $b + 4 = \infty$. This implies that $a = -8$ and $b = -\infty$. However, this is not a valid domain for the function $f(x)$, since the domain of $f(x)$ cannot be an infinite interval.

Option B: $\{x \mid 0 \ \textless \ x \ \textless \ \infty\}$

If the domain of function $g$ is $\{x \mid 0 \ \textless \ x \ \textless \ \infty\}$, then we can conclude that $a + 4 = 0$ and $b + 4 = \infty$. This implies that $a = -4$ and $b = -\infty$. This is a valid domain for the function $f(x)$, since the domain of $f(x)$ can be an infinite interval.

Conclusion

Based on the above analysis, we can conclude that the domain of function $g$ is $\{x \mid 0 \ \textless \ x \ \textless \ \infty\}$.

Final Answer

Q&A: Domain of a Composite Function

In the previous article, we discussed the domain of a composite function $g(x) = 2f(x - 4) + 3$. We analyzed the relationship between the two functions involved and determined that the domain of function $g$ is $\{x \mid 0 \ \textless \ x \ \textless \ \infty\}$. In this article, we'll answer some frequently asked questions about the domain of a composite function.

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 the case of a composite function $g(x) = 2f(x - 4) + 3$, the domain of $g(x)$ is the same as the domain of $f(x - 4)$.

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 domain of the inner function and how it affects the domain of the outer function. In the case of a composite function $g(x) = 2f(x - 4) + 3$, you need to find the domain of $f(x - 4)$ and then determine the domain of $g(x)$.

Q: What is the relationship between the domain of $f(x)$ and the domain of $g(x)$?

A: The domain of $g(x)$ is the same as the domain of $f(x - 4)$. This means that the domain of $g(x)$ is the set of all possible input values for which the function $f(x - 4)$ is defined.

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

A: Yes, the domain of a composite function can be an infinite interval. For example, if the domain of $f(x)$ is $\{x \mid 0 \ \textless \ x \ \textless \ \infty\}$, then the domain of $g(x) = 2f(x - 4) + 3$ is also $\{x \mid 0 \ \textless \ x \ \textless \ \infty\}$.

Q: How do I determine the domain of a composite function when the domain of $f(x)$ is not given?

A: If the domain of $f(x)$ is not given, you can use the fact that the domain of $g(x)$ is the same as the domain of $f(x - 4)$. This means that you can find the domain of $g(x)$ by finding the domain of $f(x - 4)$.

Q: Can the domain of a composite function be a finite interval?

A: Yes, the domain of a composite function can be a finite interval. For example, if the domain of $f(x)$ is $\{x \mid 0 \ \textless \ x \ \textless \ 4\}$, then the domain of $g(x) = 2f(x - 4) + 3$ is also $\{x \mid 4 \ \textless \ x \ \textless \ 8\}$.

Conclusion

In this article, we answered some frequently asked questions about the domain of a composite function. We discussed the relationship between the domain of $f(x)$ and the domain of $g(x)$, and we provided examples of how to find the domain of a composite function when the domain of $f(x)$ is given or not given.

Final Answer

The final answer is $\boxed{B}$.