State The Domain Of F ( X ) = X − 4 F(x) = \sqrt{x - 4} F ( X ) = X − 4 .A. X ≤ 4 X \leq 4 X ≤ 4 B. X ≥ − 4 X \geq -4 X ≥ − 4 C. X ≤ − 4 X \leq -4 X ≤ − 4 D. X ≥ 4 X \geq 4 X ≥ 4

Mar 13, 2025 by ADMIN 181 views

**Domain of a Square Root Function**

When dealing with square root functions, it's essential to consider the domain of the function. The domain of a function is the set of all possible input values for which the function is defined. In the case of the square root function, the input value must be non-negative, as the square root of a negative number is undefined in the real number system.

Understanding the Square Root Function

The given function is $f(x) = \sqrt{x - 4}$ . To find the domain of this function, we need to consider the values of $x$ for which the expression inside the square root is non-negative.

Non-Negative Expression

For the expression $x - 4$ to be non-negative, we need to find the values of $x$ that satisfy the inequality $x - 4 \geq 0$ . Solving this inequality, we get:

$x - 4 \geq 0$ $x \geq 4$

Domain of the Function

Since the expression inside the square root must be non-negative, the domain of the function $f(x) = \sqrt{x - 4}$ is the set of all values of $x$ that satisfy the inequality $x \geq 4$ . This means that the domain of the function is $x \geq 4$ .

Conclusion

In conclusion, the domain of the function $f(x) = \sqrt{x - 4}$ is $x \geq 4$ . This is because the expression inside the square root must be non-negative, and the inequality $x \geq 4$ ensures that this condition is met.

Answer

The correct answer is:

D. $x \geq 4$

Why Not the Other Options?

Let's consider why the other options are not correct:

Option A: $x \leq 4$ is incorrect because the expression inside the square root would be negative, and the square root of a negative number is undefined.
Option B: $x \geq -4$ is incorrect because the expression inside the square root would be negative for values of $x$ less than 4, and the square root of a negative number is undefined.
Option C: $x \leq -4$ is incorrect because the expression inside the square root would be negative for all values of $x$ , and the square root of a negative number is undefined.

Final Thoughts

In the previous article, we discussed the domain of the function $f(x) = \sqrt{x - 4}$ . We found that the domain of the function is $x \geq 4$ . In this article, we will answer some frequently asked questions about the domain of a square root function.

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 input value must be non-negative, as the square root of a negative number is undefined in the real number system.

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 consider the values of the input variable that make the expression inside the square root non-negative. You can do this by solving the inequality $x - a \geq 0$ , where $a$ is the constant inside the square root.

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. The range of a function is the set of all possible output values that the function can produce. For example, the domain of the function $f(x) = \sqrt{x - 4}$ is $x \geq 4$ , while the range of the function is $y \geq 0$ .

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

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

Q: Can the domain of a square root function be an interval?

A: Yes, the domain of a square root function can be an interval. For example, the function $f(x) = \sqrt{x - 4}$ has a domain of $x \geq 4$ , which is an interval.

Q: How do I determine if a square root function is defined for a given input value?

A: To determine if a square root function is defined for a given input value, you need to check if the expression inside the square root is non-negative. If the expression is non-negative, then the function is defined for that input value.

Q: What happens if the expression inside the square root is negative?

A: If the expression inside the square root is negative, then the function is undefined for that input value. This is because the square root of a negative number is undefined in the real number system.

Q: Can a square root function have a domain that includes negative numbers?

A: No, a square root function cannot have a domain that includes negative numbers. This is because the square root of a negative number is undefined in the real number system.

Q: Can a square root function have a domain that includes complex numbers?

A: Yes, a square root function can have a domain that includes complex numbers. However, this is a more advanced topic in mathematics and is typically studied in college-level courses.

Conclusion

In conclusion, the domain of a square root function is the set of all possible input values for which the function is defined. The domain of a square root function can be a single value, an interval, or a set of values. To determine the domain of a square root function, you need to consider the values of the input variable that make the expression inside the square root non-negative.