If $f(x)=\sqrt{x-3}$, Which Inequality Can Be Used To Find The Domain Of $f(x$\]?A. $\sqrt{x-3} \geq 0$B. $x-3 \geq 0$C. $\sqrt{x-3} \leq 0$D. $x-3 \leq 0$

Feb 28, 2025 by ADMIN 156 views

**Understanding the Domain of a Square Root Function**

When dealing with square root functions, it's essential to understand the concept of the domain. The domain of a function is the set of all possible input values for which the function is defined. In the case of a square root function, the domain is restricted to non-negative values under the square root sign.

The Square Root Function

The given function is $f(x) = \sqrt{x-3}$ . To find the domain of this function, we need to determine the values of $x$ for which the expression under the square root sign, $x-3$ , is non-negative.

Why Non-Negativity is Crucial

The square root of a negative number is undefined in the real number system. Therefore, to ensure that the function is defined, we must have $x-3 \geq 0$ . This is because the square root of a non-negative number is always non-negative.

Solving the Inequality

To find the values of $x$ that satisfy the inequality $x-3 \geq 0$ , we can add $3$ to both sides of the inequality. This gives us $x \geq 3$ .

Conclusion

The inequality that can be used to find the domain of $f(x)$ is $x-3 \geq 0$ . This inequality ensures that the expression under the square root sign is non-negative, making the function defined for all values of $x$ greater than or equal to $3$ .

Answer

The correct answer is B. $x-3 \geq 0$ .

Additional Insights

It's worth noting that the other options are incorrect because they do not ensure that the expression under the square root sign is non-negative. Option A, $\sqrt{x-3} \geq 0$ , is incorrect because it's the square root that's non-negative, not the expression under the square root sign. Option C, $\sqrt{x-3} \leq 0$ , is also incorrect because the square root of a non-negative number is always non-negative. Option D, $x-3 \leq 0$ , is incorrect because it restricts the domain to values less than $3$ , which is not the correct domain for the function.

Real-World Applications

Understanding the domain of a function is crucial in various real-world applications, such as:

Physics: When dealing with physical systems, it's essential to ensure that the input values are within the domain of the function to avoid undefined or imaginary results.
Engineering: In engineering applications, the domain of a function can affect the accuracy and reliability of the results.
Computer Science: In computer science, understanding the domain of a function is crucial when working with algorithms and data structures.

Conclusion

Q: What is the domain of a square root function?

A: The domain of a square root function is the set of all possible input values for which the function is defined. In the case of a square root function, the domain is restricted to non-negative values under the square root sign.

Q: Why is it essential to ensure that the expression under the square root sign is non-negative?

A: The square root of a negative number is undefined in the real number system. Therefore, to ensure that the function is defined, we must have the expression under the square root sign non-negative.

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

A: To find the domain of a square root function, you need to determine the values of the input variable for which the expression under the square root sign is non-negative. This can be done by solving the inequality $x-3 \geq 0$ , where $x$ is the input variable.

Q: What is the correct inequality to use when finding the domain of a square root function?

A: The correct inequality to use when finding the domain of a square root function is $x-3 \geq 0$ . This ensures that the expression under the square root sign is non-negative, making the function defined for all values of $x$ greater than or equal to $3$ .

Q: What are some real-world applications of understanding the domain of a square root function?

A: Understanding the domain of a square root function is crucial in various real-world applications, such as:

Physics: When dealing with physical systems, it's essential to ensure that the input values are within the domain of the function to avoid undefined or imaginary results.
Engineering: In engineering applications, the domain of a function can affect the accuracy and reliability of the results.
Computer Science: In computer science, understanding the domain of a function is crucial when working with algorithms and data structures.

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

A: To determine the domain of a square root function with a constant under the square root sign, you need to find the value of the constant that makes the expression under the square root sign non-negative. For example, if the function is $f(x) = \sqrt{4}$ , the domain is all real numbers, since $4$ is a non-negative constant.

Q: Can I have a negative value under the square root sign?

A: No, you cannot have a negative value under the square root sign. The square root of a negative number is undefined in the real number system.

Q: What happens if I have a negative value under the square root sign?

A: If you have a negative value under the square root sign, the function is undefined for that value. This is because the square root of a negative number is undefined in the real number system.

Q: Can I use a complex number under the square root sign?

A: Yes, you can use a complex number under the square root sign. However, the result will be a complex number, not a real number.

Q: What is the difference between a square root function and a cube root function?

A: The main difference between a square root function and a cube root function is the power under the root sign. A square root function has a power of $2$ , while a cube root function has a power of $3$ . This affects the domain and range of the functions.

Q: Can I have a fractional power under the root sign?

A: Yes, you can have a fractional power under the root sign. However, the result will be a root that is not a whole number.

Q: What are some common mistakes to avoid when working with square root functions?

A: Some common mistakes to avoid when working with square root functions include:

Not checking the domain: Make sure to check the domain of the function to avoid undefined or imaginary results.
Not simplifying the expression: Simplify the expression under the square root sign to make it easier to work with.
Not using the correct inequality: Use the correct inequality to find the domain of the function.
Not considering complex numbers: Consider complex numbers when working with square root functions.

Conclusion

In conclusion, understanding the domain of a square root function is crucial in various real-world applications. By following the correct steps and avoiding common mistakes, you can ensure that your results are accurate and reliable.