What Is The Domain Of The Function G ( X ) = 3 Log ⁡ 2 ( X − 1 ) + 4 G(x) = 3 \log_2(x-1) + 4 G ( X ) = 3 Lo G 2 ​ ( X − 1 ) + 4 ?Enter Your Answer In The Box.All Real Numbers Greater Than { \square$}$.

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Understanding the Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined and produces a real number as output. In other words, it is the set of all possible values of x that can be plugged into the function without causing any problems or resulting in an undefined value.

The Logarithmic Function

The function $g(x) = 3 \log_2(x-1) + 4$ involves a logarithmic function, specifically the base-2 logarithm. The logarithmic function is defined only for positive real numbers, and its domain is typically represented as $(0, \infty)$.

The Domain of the Function $g(x)$

To find the domain of the function $g(x)$, we need to consider the restrictions imposed by the logarithmic function. Since the logarithm of a non-positive number is undefined, we must ensure that the argument of the logarithm, $x-1$, is always positive.

Restriction on the Argument of the Logarithm

The argument of the logarithm, $x-1$, must be greater than 0. This means that $x-1 > 0$, which can be rewritten as $x > 1$. Therefore, the domain of the function $g(x)$ is all real numbers greater than 1.

Conclusion

In conclusion, the domain of the function $g(x) = 3 \log_2(x-1) + 4$ is all real numbers greater than 1. This is because the logarithmic function is defined only for positive real numbers, and the argument of the logarithm, $x-1$, must be greater than 0.

Final Answer

The final answer is $\boxed{1}$.

Discussion

The domain of a function is an essential concept in mathematics, and it plays a crucial role in understanding the behavior and properties of functions. In this discussion, we have explored the domain of the function $g(x) = 3 \log_2(x-1) + 4$ and found that it is all real numbers greater than 1.

Related Topics

• Domain of a function
• Logarithmic function
• Restrictions on the argument of a logarithm

Example Problems

• Find the domain of the function $f(x) = 2 \log_3(x+2) - 1$.
• Determine the domain of the function $h(x) = \log_5(x-3) + 2$.

Solutions to Example Problems

• The domain of the function $f(x) = 2 \log_3(x+2) - 1$ is all real numbers greater than -2.
• The domain of the function $h(x) = \log_5(x-3) + 2$ is all real numbers greater than 3.

Tips and Tricks

• When dealing with logarithmic functions, always ensure that the argument of the logarithm is positive.
• The domain of a function can be restricted by the properties of the function itself, such as the logarithmic function.
• Understanding the domain of a function is essential for analyzing and working with functions in mathematics.
  

Understanding the Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined and produces a real number as output. In this article, we will explore the domain of a function and answer some frequently asked questions related to this topic.

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values (x-values) for which the function is defined and produces a real number as output.

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

A: To find the domain of a function, you need to consider the restrictions imposed by the function itself. For example, if the function involves a logarithmic function, you need to ensure that the argument of the logarithm is positive.

Q: What is the domain of the function $f(x) = \log_2(x)$?

A: The domain of the function $f(x) = \log_2(x)$ is all real numbers greater than 0.

Q: What is the domain of the function $g(x) = 3 \log_2(x-1) + 4$?

A: The domain of the function $g(x) = 3 \log_2(x-1) + 4$ is all real numbers greater than 1.

Q: How do I determine the domain of a function with a square root?

A: To determine the domain of a function with a square root, you need to ensure that the argument of the square root is non-negative.

Q: What is the domain of the function $h(x) = \sqrt{x-4}$?

A: The domain of the function $h(x) = \sqrt{x-4}$ is all real numbers greater than or equal to 4.

Q: How do I find the domain of a function with a rational expression?

A: To find the domain of a function with a rational expression, you need to ensure that the denominator is not equal to zero.

Q: What is the domain of the function $f(x) = \frac{1}{x-2}$?

A: The domain of the function $f(x) = \frac{1}{x-2}$ is all real numbers except 2.

Q: Can the domain of a function be empty?

A: Yes, the domain of a function can be empty. For example, the function $f(x) = \frac{1}{0}$ has an empty domain.

Q: How do I determine the domain of a function with a trigonometric function?

A: To determine the domain of a function with a trigonometric function, you need to consider the restrictions imposed by the function itself. For example, the sine function is defined for all real numbers, but the cosine function is not defined for odd multiples of $\frac{\pi}{2}$.

Q: What is the domain of the function $g(x) = \sin(x)$?

A: The domain of the function $g(x) = \sin(x)$ is all real numbers.

Q: What is the domain of the function $h(x) = \cos(x)$?

A: The domain of the function $h(x) = \cos(x)$ is all real numbers except odd multiples of $\frac{\pi}{2}$.

Q: Can the domain of a function be a single value?

A: Yes, the domain of a function can be a single value. For example, the function $f(x) = \sqrt{x}$ has a domain of 0.

Q: How do I determine the domain of a function with a piecewise function?

A: To determine the domain of a function with a piecewise function, you need to consider the restrictions imposed by each piece of the function.

Q: What is the domain of the function $f(x) = \begin{cases} x & \text{if } x < 0 \\ 1 & \text{if } x \geq 0 \end{cases}$?

A: The domain of the function $f(x) = \begin{cases} x & \text{if } x < 0 \\ 1 & \text{if } x \geq 0 \end{cases}$ is all real numbers.

Q: Can the domain of a function be a union of intervals?

A: Yes, the domain of a function can be a union of intervals. For example, the function $f(x) = \begin{cases} x & \text{if } x < 0 \\ 1 & \text{if } 0 \leq x < 1 \\ 2 & \text{if } x \geq 1 \end{cases}$ has a domain of $(-\infty, 0) \cup [1, \infty)$.

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

A: To determine the domain of a function with a composite function, you need to consider the restrictions imposed by each function in the composition.

Q: What is the domain of the function $f(x) = \sin(\cos(x))$?

A: The domain of the function $f(x) = \sin(\cos(x))$ is all real numbers.

Q: Can the domain of a function be a combination of intervals and single values?

A: Yes, the domain of a function can be a combination of intervals and single values. For example, the function $f(x) = \begin{cases} x & \text{if } x < 0 \\ 1 & \text{if } x = 0 \\ 2 & \text{if } x > 0 \end{cases}$ has a domain of $(-\infty, 0) \cup \{0\} \cup (0, \infty)$.

Q: How do I determine the domain of a function with a parametric function?

A: To determine the domain of a function with a parametric function, you need to consider the restrictions imposed by the parameter.

Q: What is the domain of the function $f(x) = \begin{cases} x & \text{if } x < 0 \\ 1 & \text{if } x \geq 0 \end{cases}$?

A: The domain of the function $f(x) = \begin{cases} x & \text{if } x < 0 \\ 1 & \text{if } x \geq 0 \end{cases}$ is all real numbers.

Q: Can the domain of a function be a combination of intervals, single values, and parametric functions?

A: Yes, the domain of a function can be a combination of intervals, single values, and parametric functions. For example, the function $f(x) = \begin{cases} x & \text{if } x < 0 \\ 1 & \text{if } x = 0 \\ 2 & \text{if } x > 0 \\ \sin(t) & \text{if } t \in [0, 2\pi] \end{cases}$ has a domain of $(-\infty, 0) \cup \{0\} \cup (0, \infty) \cup [0, 2\pi]$.

Q: How do I determine the domain of a function with a multivariable function?

A: To determine the domain of a function with a multivariable function, you need to consider the restrictions imposed by each variable.

Q: What is the domain of the function $f(x, y) = \frac{1}{x+y}$?

A: The domain of the function $f(x, y) = \frac{1}{x+y}$ is all real numbers except $x+y = 0$.

Q: Can the domain of a function be a combination of intervals, single values, parametric functions, and multivariable functions?

A: Yes, the domain of a function can be a combination of intervals, single values, parametric functions, and multivariable functions. For example, the function $f(x, y) = \begin{cases} x & \text{if } x < 0 \\ 1 & \text{if } x = 0 \\ 2 & \text{if } x > 0 \\ \sin(t) & \text{if } t \in [0, 2\pi] \\ \frac{1}{x+y} & \text{if } x+y \neq 0 \end{cases}$ has a domain of $(-\infty, 0) \cup \{0\} \cup (0, \infty) \cup [0, 2\pi] \cup \{(x, y) \in \mathbb{R}^2 : x+y \neq 0\}$.

Q: How do I determine the domain of a function with a complex function?

A: To determine the domain of a function with a complex function, you need to consider the restrictions imposed by the complex function.

Q: What is the domain of the function $f(z) = \frac{1}{z}$?

A: The domain of the function $f(z) = \frac{1}{z}$ is all complex numbers except $z = 0$.

Q: Can the domain of a function be a combination of intervals, single values, parametric functions, multivariable functions, and complex functions?

A: Yes, the domain of a function can be a combination of intervals, single values, parametric functions, multivariable functions, and