Find The Domain Of The Following Function:${ F(x) = \frac{1}{\sqrt{4x - 12}} }$Give Your Answer In Interval Notation.Provide Your Answer Below:

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Introduction

In mathematics, the domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible values of x for which the function f(x) is a real number. In this article, we will find the domain of the function $f(x) = \frac{1}{\sqrt{4x - 12}}$.

Understanding the Function

The given function is $f(x) = \frac{1}{\sqrt{4x - 12}}$. This function has a square root in the denominator, which means that the expression inside the square root must be non-negative. In other words, $4x - 12 \geq 0$. We will use this inequality to find the domain of the function.

Finding the Domain

To find the domain of the function, we need to solve the inequality $4x - 12 \geq 0$. We can start by adding 12 to both sides of the inequality, which gives us $4x \geq 12$. Next, we can divide both sides of the inequality by 4, which gives us $x \geq 3$.

However, we also need to consider the fact that the expression inside the square root must be non-negative. In other words, $4x - 12 \geq 0$. We can rewrite this inequality as $x \geq 3$. Therefore, the domain of the function is $x \geq 3$.

Writing the Domain in Interval Notation

The domain of the function is $x \geq 3$, which can be written in interval notation as $[3, \infty)$.

Conclusion

In this article, we found the domain of the function $f(x) = \frac{1}{\sqrt{4x - 12}}$. We used the inequality $4x - 12 \geq 0$ to find the domain of the function, and we wrote the domain in interval notation as $[3, \infty)$.

Step-by-Step Solution

Here is a step-by-step solution to find the domain of the function:

1. Start with the inequality $4x - 12 \geq 0$.
2. Add 12 to both sides of the inequality to get $4x \geq 12$.
3. Divide both sides of the inequality by 4 to get $x \geq 3$.
4. Consider the fact that the expression inside the square root must be non-negative.
5. Rewrite the inequality as $x \geq 3$.
6. Write the domain of the function in interval notation as $[3, \infty)$.

Example

Here is an example of how to find the domain of a function:

Find the domain of the function $f(x) = \frac{1}{\sqrt{x - 2}}$.

Solution:

1. Start with the inequality $x - 2 \geq 0$.
2. Add 2 to both sides of the inequality to get $x \geq 2$.
3. Consider the fact that the expression inside the square root must be non-negative.
4. Rewrite the inequality as $x \geq 2$.
5. Write the domain of the function in interval notation as $[2, \infty)$.

Tips and Tricks

Here are some tips and tricks to help you find the domain of a function:

• Always start with the inequality that is inside the square root.
• Add or subtract the same value to both sides of the inequality to isolate the variable.
• Consider the fact that the expression inside the square root must be non-negative.
• Rewrite the inequality as a statement about the variable.
• Write the domain of the function in interval notation.

Common Mistakes

Here are some common mistakes to avoid when finding the domain of a function:

• Failing to consider the fact that the expression inside the square root must be non-negative.
• Not rewriting the inequality as a statement about the variable.
• Not writing the domain of the function in interval notation.

Real-World Applications

Here are some real-world applications of finding the domain of a function:

• In physics, the domain of a function can represent the range of possible values for a physical quantity.
• In engineering, the domain of a function can represent the range of possible values for a design parameter.
• In economics, the domain of a function can represent the range of possible values for a economic variable.

Final Answer

The final answer is $\boxed{[3, \infty)}$.

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Introduction

In our previous article, we found the domain of the function $f(x) = \frac{1}{\sqrt{4x - 12}}$. In this article, we will answer some common questions related to finding the domain of a function.

Q&A

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible values of x for which the function f(x) is a real number.

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

A: To find the domain of a function, you need to solve the inequality that is inside the square root. You can start by adding or subtracting the same value to both sides of the inequality to isolate the variable. Then, consider the fact that the expression inside the square root must be non-negative. Finally, rewrite the inequality as a statement about the variable and write the domain of the function in interval notation.

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

A: The domain of a function is the set of all possible input values for which the function is defined, while the range of a function is the set of all possible output values. In other words, the domain is the set of all possible values of x, while the range is the set of all possible values of f(x).

Q: How do I write the domain of a function in interval notation?

A: To write the domain of a function in interval notation, you need to use the following notation:

• [a, b] represents the set of all real numbers between a and b, including a and b.
• (a, b) represents the set of all real numbers between a and b, excluding a and b.
• [a, ∞) represents the set of all real numbers greater than or equal to a.
• (-∞, a) represents the set of all real numbers less than or equal to a.

Q: What are some common mistakes to avoid when finding the domain of a function?

A: Some common mistakes to avoid when finding the domain of a function include:

• Failing to consider the fact that the expression inside the square root must be non-negative.
• Not rewriting the inequality as a statement about the variable.
• Not writing the domain of the function in interval notation.

Q: How do I use the domain of a function in real-world applications?

A: The domain of a function can be used in real-world applications in a variety of ways, including:

• In physics, the domain of a function can represent the range of possible values for a physical quantity.
• In engineering, the domain of a function can represent the range of possible values for a design parameter.
• In economics, the domain of a function can represent the range of possible values for an economic variable.

Example Questions

Here are some example questions related to finding the domain of a function:

• Find the domain of the function $f(x) = \frac{1}{\sqrt{x - 2}}$.
• Find the domain of the function $f(x) = \frac{1}{\sqrt{2x - 5}}$.
• Find the domain of the function $f(x) = \frac{1}{\sqrt{x + 3}}$.

Solutions

Here are the solutions to the example questions:

• The domain of the function $f(x) = \frac{1}{\sqrt{x - 2}}$ is $[2, \infty)$.
• The domain of the function $f(x) = \frac{1}{\sqrt{2x - 5}}$ is $[\frac{5}{2}, \infty)$.
• The domain of the function $f(x) = \frac{1}{\sqrt{x + 3}}$ is $[-3, \infty)$.

Tips and Tricks

Here are some tips and tricks to help you find the domain of a function:

• Always start with the inequality that is inside the square root.
• Add or subtract the same value to both sides of the inequality to isolate the variable.
• Consider the fact that the expression inside the square root must be non-negative.
• Rewrite the inequality as a statement about the variable.
• Write the domain of the function in interval notation.

Common Mistakes

Here are some common mistakes to avoid when finding the domain of a function:

• Failing to consider the fact that the expression inside the square root must be non-negative.
• Not rewriting the inequality as a statement about the variable.
• Not writing the domain of the function in interval notation.

Real-World Applications

Here are some real-world applications of finding the domain of a function:

• In physics, the domain of a function can represent the range of possible values for a physical quantity.
• In engineering, the domain of a function can represent the range of possible values for a design parameter.
• In economics, the domain of a function can represent the range of possible values for an economic variable.

Final Answer

The final answer is $\boxed{[3, \infty)}$.