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

Feb 28, 2025 by ADMIN 214 views

Domain of a Composite Function: Understanding the Function g(x)

When dealing with composite functions, it's essential to understand how the domain of the original function affects the domain of the composite function. In this article, we'll explore the domain of the function g(x), which is defined as g(x) = 2f(x-4) + 3.

Understanding the Function g(x)

The function g(x) is a composite function, meaning it's a combination of two or more functions. In this case, g(x) is a combination of the function f(x) and a constant value of 3. The function f(x) is not explicitly defined in the problem, but we can still analyze the domain of g(x) based on the given information.

Analyzing the Domain of g(x)

To find the domain of g(x), we need to consider the domain of the function f(x). Since f(x) is not explicitly defined, we'll assume that it has a domain of all real numbers, denoted as (-∞, ∞). This means that f(x) is defined for all values of x.

Now, let's analyze the function g(x) = 2f(x-4) + 3. The function f(x-4) is a shifted version of the function f(x), where the input x is replaced by x-4. This means that the domain of f(x-4) is also all real numbers, denoted as (-∞, ∞).

However, we need to consider the fact that the input x-4 is used in the function f(x-4). This means that the domain of g(x) is restricted to values of x that make x-4 a valid input for the function f(x-4).

Restricting the Domain of g(x)

To find the valid values of x, we need to consider the domain of the function f(x). Since f(x) is defined for all real numbers, the function f(x-4) is also defined for all real numbers. However, we need to consider the fact that the input x-4 is used in the function f(x-4).

Let's assume that the function f(x) has a domain of [a, b], where a and b are real numbers. Then, the function f(x-4) has a domain of [a+4, b+4]. This means that the function g(x) = 2f(x-4) + 3 has a domain of [a+4, b+4].

Finding the Domain of g(x)

Since the function f(x) is not explicitly defined, we'll assume that it has a domain of all real numbers, denoted as (-∞, ∞). This means that the function f(x-4) also has a domain of all real numbers, denoted as (-∞, ∞).

Let's consider the inequality x-4 ≥ a, where a is a real number. This inequality represents the condition that x-4 must be greater than or equal to a. Solving for x, we get x ≥ a+4.

Similarly, let's consider the inequality x-4 ≤ b, where b is a real number. This inequality represents the condition that x-4 must be less than or equal to b. Solving for x, we get x ≤ b+4.

Combining the Inequalities

To find the domain of g(x), we need to combine the inequalities x ≥ a+4 and x ≤ b+4. This means that the domain of g(x) is the intersection of the two intervals [a+4, ∞) and (-∞, b+4].

Simplifying the Domain

Since the function f(x) has a domain of all real numbers, the domain of g(x) is also all real numbers. However, we need to consider the fact that the input x-4 is used in the function f(x-4).

Let's consider the inequality x-4 ≥ -4. This inequality represents the condition that x-4 must be greater than or equal to -4. Solving for x, we get x ≥ 0.

Similarly, let's consider the inequality x-4 ≤ ∞. This inequality represents the condition that x-4 must be less than or equal to ∞. Solving for x, we get x ≤ ∞.

Finding the Domain of g(x)

To find the domain of g(x), we need to combine the inequalities x ≥ 0 and x ≤ ∞. This means that the domain of g(x) is the interval [0, ∞).

Conclusion

In conclusion, the domain of the function g(x) = 2f(x-4) + 3 is the interval [0, ∞). This means that the function g(x) is defined for all values of x that are greater than or equal to 0.

Answer

The correct answer is A. {x | 0 ≤ x < ∞}.
Domain of a Composite Function: Understanding the Function g(x) - Q&A

In our previous article, we explored the domain of the function g(x), which is defined as g(x) = 2f(x-4) + 3. We found that the domain of g(x) is the interval [0, ∞). In this article, we'll answer some frequently asked questions about the domain of g(x).

Q: What is the domain of the function f(x)?

A: The domain of the function f(x) is not explicitly defined in the problem. However, we can assume that it has a domain of all real numbers, denoted as (-∞, ∞).

Q: How does the domain of f(x) affect the domain of g(x)?

A: The domain of g(x) is restricted to values of x that make x-4 a valid input for the function f(x-4). This means that the domain of g(x) is the intersection of the two intervals [a+4, ∞) and (-∞, b+4), where a and b are real numbers.

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

A: The domain of g(x) is a subset of the domain of f(x). This means that the domain of g(x) is restricted to values of x that make x-4 a valid input for the function f(x-4).

Q: Can the domain of g(x) be different from the domain of f(x)?

A: Yes, the domain of g(x) can be different from the domain of f(x). This is because the function g(x) is a composite function, and the domain of g(x) is restricted to values of x that make x-4 a valid input for the function f(x-4).

Q: How do I find the domain of g(x) if the domain of f(x) is not explicitly defined?

A: If the domain of f(x) is not explicitly defined, you can assume that it has a domain of all real numbers, denoted as (-∞, ∞). Then, you can find the domain of g(x) by considering the inequality x-4 ≥ a, where a is a real number. Solving for x, you get x ≥ a+4.

Q: Can the domain of g(x) be empty?

A: No, the domain of g(x) cannot be empty. This is because the function g(x) is defined for all values of x that make x-4 a valid input for the function f(x-4).

Q: How do I determine if the domain of g(x) is bounded or unbounded?

A: To determine if the domain of g(x) is bounded or unbounded, you need to consider the inequality x-4 ≤ b, where b is a real number. Solving for x, you get x ≤ b+4. If b is finite, then the domain of g(x) is bounded. If b is infinite, then the domain of g(x) is unbounded.

Q: Can the domain of g(x) be a single point?

A: No, the domain of g(x) cannot be a single point. This is because the function g(x) is defined for all values of x that make x-4 a valid input for the function f(x-4).

Conclusion

In conclusion, the domain of the function g(x) = 2f(x-4) + 3 is the interval [0, ∞). We hope that this Q&A article has helped you understand the domain of g(x) and how it relates to the domain of the function f(x). If you have any further questions, please don't hesitate to ask.

Answer

The correct answer is A. {x | 0 ≤ x < ∞}.