Find The Domain Of The Function F ( X ) = 3 X − 7 F(x) = \sqrt{3x - 7} F ( X ) = 3 X − 7 ​ .In Interval Form, The Domain Of F F F Is:

Introduction

In mathematics, a function is a relation between a set of inputs, called the domain, and a set of possible outputs. The domain of a function is the set of all possible input values for which the function is defined. In this article, we will focus on finding the domain of the function $f(x) = \sqrt{3x - 7}$.

What is the Domain of a Function?

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 defined. For example, consider the function $f(x) = \frac{1}{x}$. The domain of this function is all real numbers except $0$, because division by zero is undefined.

Finding the Domain of a Function

To find the domain of a function, we need to identify the values of $x$ for which the function is defined. In the case of the function $f(x) = \sqrt{3x - 7}$, we need to find the values of $x$ for which the expression inside the square root is non-negative.

The Square Root Function

The square root function is defined as $\sqrt{x} = y$ if and only if $x = y^2$. In other words, the square root function returns the value of $y$ such that $y^2 = x$. The domain of the square root function is all non-negative real numbers.

Finding the Domain of $f(x) = \sqrt{3x - 7}$

To find the domain of the function $f(x) = \sqrt{3x - 7}$, we need to find the values of $x$ for which the expression inside the square root is non-negative. In other words, we need to find the values of $x$ for which $3x - 7 \geq 0$.

Solving the Inequality

To solve the inequality $3x - 7 \geq 0$, we can add $7$ to both sides of the inequality to get $3x \geq 7$. Then, we can divide both sides of the inequality by $3$ to get $x \geq \frac{7}{3}$.

Conclusion

In conclusion, the domain of the function $f(x) = \sqrt{3x - 7}$ is all real numbers greater than or equal to $\frac{7}{3}$. In interval form, the domain of $f$ is $[\frac{7}{3}, \infty)$.

Example Problems

Here are some example problems to help you practice finding the domain of a function:

• Find the domain of the function $f(x) = \sqrt{2x + 5}$.
• Find the domain of the function $f(x) = \sqrt{x - 2}$.
• Find the domain of the function $f(x) = \sqrt{4x - 9}$.

Step-by-Step Solutions

Here are the step-by-step solutions to the example problems:

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

  1. The expression inside the square root is $2x + 5$.
  2. We need to find the values of $x$ for which $2x + 5 \geq 0$.
  3. Subtracting $5$ from both sides of the inequality, we get $2x \geq -5$.
  4. Dividing both sides of the inequality by $2$, we get $x \geq -\frac{5}{2}$.
  5. Therefore, the domain of the function $f(x) = \sqrt{2x + 5}$ is all real numbers greater than or equal to $-\frac{5}{2}$.

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

  1. The expression inside the square root is $x - 2$.
  2. We need to find the values of $x$ for which $x - 2 \geq 0$.
  3. Adding $2$ to both sides of the inequality, we get $x \geq 2$.
  4. Therefore, the domain of the function $f(x) = \sqrt{x - 2}$ is all real numbers greater than or equal to $2$.

• Find the domain of the function $f(x) = \sqrt{4x - 9}$.

  1. The expression inside the square root is $4x - 9$.
  2. We need to find the values of $x$ for which $4x - 9 \geq 0$.
  3. Adding $9$ to both sides of the inequality, we get $4x \geq 9$.
  4. Dividing both sides of the inequality by $4$, we get $x \geq \frac{9}{4}$.
  5. Therefore, the domain of the function $f(x) = \sqrt{4x - 9}$ is all real numbers greater than or equal to $\frac{9}{4}$.

Final Thoughts

Frequently Asked Questions

In this article, we will answer some frequently asked questions about the domain of a function.

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.

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

A: To find the domain of a function, you need to identify the values of $x$ for which the function is defined. In the case of the function $f(x) = \sqrt{3x - 7}$, you need to find the values of $x$ for which the expression inside the square root is non-negative.

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 of the function.

Q: Can a function have an empty domain?

A: Yes, a function can have an empty domain. For example, the function $f(x) = \frac{1}{x}$ has an empty domain because division by zero is undefined.

Q: Can a function have a domain that is all real numbers?

A: Yes, a function can have a domain that is all real numbers. For example, the function $f(x) = x^2$ has a domain that is all real numbers.

Q: How do I determine if a function is defined at a particular point?

A: To determine if a function is defined at a particular point, you need to check if the function is defined at that point. For example, the function $f(x) = \frac{1}{x}$ is not defined at $x = 0$ because division by zero is undefined.

Q: Can a function have multiple domains?

A: No, a function cannot have multiple domains. The domain of a function is a set of all possible input values for which the function is defined, and it is unique for each function.

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

A: To find the domain of a function with a square root, you need to find the values of $x$ for which the expression inside the square root is non-negative.

Q: Can a function with a square root have a domain that is all real numbers?

A: No, a function with a square root cannot have a domain that is all real numbers. The expression inside the square root must be non-negative for the function to be defined.

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

A: To find the domain of a function with a fraction, you need to find the values of $x$ for which the denominator is not equal to zero.

Q: Can a function with a fraction have a domain that is all real numbers?

A: No, a function with a fraction cannot have a domain that is all real numbers. The denominator must be non-zero for the function to be defined.

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

A: To find the domain of a function with a polynomial, you need to find the values of $x$ for which the polynomial is defined.

Q: Can a function with a polynomial have a domain that is all real numbers?

A: Yes, a function with a polynomial can have a domain that is all real numbers.

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

A: To find the domain of a function with a trigonometric function, you need to find the values of $x$ for which the trigonometric function is defined.

Q: Can a function with a trigonometric function have a domain that is all real numbers?

A: Yes, a function with a trigonometric function can have a domain that is all real numbers.

Conclusion

In conclusion, the domain of a function is an important concept in mathematics. By understanding the concept of the domain of a function, we can solve a wide range of problems in mathematics and other fields. In this article, we have answered some frequently asked questions about the domain of a function. We hope that this article has been helpful in understanding the concept of the domain of a function.