Determining the Domain of Function g

Oct 13, 2025 by miodeya.com 128 views

Introduction

In mathematics, functions are used to describe relationships between variables. When dealing with functions, it's essential to understand their domains, which represent the set of all possible input values for which the function is defined. In this article, we will explore the domain of function g, which is defined as g(x) = 2f(x - 4) + 3, where f(x) = log2 x.

Understanding the Function f(x)

The function f(x) = log2 x is a logarithmic function with base 2. This function is defined for all positive real numbers, and its range is all real numbers. The graph of f(x) is a curve that increases as x increases, with a vertical asymptote at x = 0.

Defining Function g(x)

Function g(x) is defined as g(x) = 2f(x - 4) + 3. This function is a transformation of the function f(x), where the input x is shifted 4 units to the right, and the output is multiplied by 2 and then shifted 3 units upwards.

Determining the Domain of Function g

To determine the domain of function g, we need to consider the restrictions imposed by the function f(x). Since f(x) is defined for all positive real numbers, the input x - 4 must be positive for f(x - 4) to be defined. This implies that x - 4 > 0, or x > 4.

However, we also need to consider the fact that the logarithmic function f(x) is not defined for non-positive real numbers. Since x - 4 must be positive, we can conclude that x - 4 > 0, or x > 4.

Analyzing the Transformation

The transformation x - 4 shifts the input to the right by 4 units, which means that the domain of function g is shifted 4 units to the right compared to the domain of function f. Since the domain of function f is all positive real numbers, the domain of function g is also all positive real numbers.

Conclusion

In conclusion, the domain of function g is all positive real numbers, which can be represented as {x | x > 4}. This means that function g is defined for all x greater than 4.

Final Answer

The final answer is {x | x > 4}.

References

[1] "Logarithmic Functions" by Math Open Reference
[2] "Function Transformations" by Khan Academy

Additional Resources

[1] "Domain and Range of Functions" by Math Is Fun
[2] "Logarithmic Functions" by Wolfram MathWorld
Q&A: The Domain of Function g =====================================

Introduction

In our previous article, we explored the domain of function g, which is defined as g(x) = 2f(x - 4) + 3, where f(x) = log2 x. We concluded that the domain of function g is all positive real numbers, represented as {x | x > 4}. In this article, we will answer some frequently asked questions about the domain of function g.

Q: What is the domain of function g?

A: The domain of function g is all positive real numbers, represented as {x | x > 4}.

Q: Why is the domain of function g all positive real numbers?

A: The domain of function g is all positive real numbers because the function f(x) = log2 x is defined for all positive real numbers. Since function g is a transformation of function f, the domain of function g is also all positive real numbers.

Q: What happens if x is less than or equal to 4?

A: If x is less than or equal to 4, then x - 4 is non-positive, and the function f(x - 4) is not defined. Therefore, function g is not defined for x less than or equal to 4.

Q: Can function g be defined for x less than or equal to 4?

A: No, function g cannot be defined for x less than or equal to 4. The function f(x) = log2 x is not defined for non-positive real numbers, and the transformation x - 4 shifts the input to the right by 4 units, which means that the domain of function g is also shifted 4 units to the right.

Q: What is the relationship between the domain of function f and the domain of function g?

A: The domain of function g is a transformation of the domain of function f. Since the domain of function f is all positive real numbers, the domain of function g is also all positive real numbers.

Q: Can the domain of function g be changed?

A: No, the domain of function g cannot be changed. The domain of function g is determined by the function f(x) = log2 x and the transformation x - 4.

Q: What is the significance of the domain of function g?

A: The domain of function g is significant because it determines the set of all possible input values for which the function is defined. Understanding the domain of function g is essential for working with the function and making predictions about its behavior.