Restricting the Domain of the Square Root Function

Oct 13, 2025 by windailys.com 112 views

The square root function, denoted as $f(x) = \sqrt{x}$ , is a fundamental concept in mathematics that has numerous applications in various fields. 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 domain is all non-negative real numbers, i.e., $x \geq 0$ . However, when the domain is restricted to $x \leq 7$ , we need to determine which statement must be true.

Restricting the Domain of the Square Root Function

When the domain of the square root function is restricted to $x \leq 7$ , it means that the function is only defined for values of $x$ less than or equal to 7. This restriction is often imposed to ensure that the function is well-defined and does not produce any imaginary or complex numbers.

Analyzing the Statements

Let's analyze each of the given statements to determine which one must be true when the domain of the square root function is restricted to $x \leq 7$ .

A. 7 is subtracted from the $x$ -term inside the radical

If 7 is subtracted from the $x$ -term inside the radical, the function becomes $f(x) = \sqrt{x - 7}$ . This means that the domain of the function is shifted to the right by 7 units, resulting in a new domain of $x - 7 \geq 0$ , which simplifies to $x \geq 7$ . However, this contradicts the given restriction that the domain is $x \leq 7$ . Therefore, statement A is not true.

B. The radical is multiplied by a negative number

If the radical is multiplied by a negative number, the function becomes $f(x) = -\sqrt{x}$ . This means that the function is reflected across the $x$ -axis, but it does not affect the domain of the function. The domain remains the same, i.e., $x \geq 0$ . However, when the domain is restricted to $x \leq 7$ , the function is still defined for values of $x$ less than or equal to 7. Therefore, statement B is not true.

C. 7 is added to the $x$ -term inside the radical

If 7 is added to the $x$ -term inside the radical, the function becomes $f(x) = \sqrt{x + 7}$ . This means that the domain of the function is shifted to the left by 7 units, resulting in a new domain of $x + 7 \geq 0$ , which simplifies to $x \geq -7$ . However, when the domain is restricted to $x \leq 7$ , the function is still defined for values of $x$ less than or equal to 7. Therefore, statement C is not true.

Conclusion

Based on the analysis of the statements, we can conclude that none of the statements A, B, or C must be true when the domain of the square root function is restricted to $x \leq 7$ . However, we can make a more general statement that must be true.

The Correct Statement

When the domain of the square root function is restricted to $x \leq 7$ , the statement that must be true is that the function is defined for values of $x$ less than or equal to 7. This is because the domain of the function is restricted to $x \leq 7$ , and the function is only defined for values of $x$ within this restricted domain.

Example

To illustrate this concept, let's consider an example. Suppose we have a square root function defined as $f(x) = \sqrt{x}$ , and we want to restrict the domain to $x \leq 7$ . In this case, the function is only defined for values of $x$ less than or equal to 7. For example, if we input $x = 5$ , the function evaluates to $f(5) = \sqrt{5}$ . However, if we input $x = 8$ , the function is undefined because 8 is not within the restricted domain.

Conclusion

$In conclusion, when the domain of the square root function is restricted to $x \leq 7$, the statement that must be true is that the function is defined for values of $x$ less than or equal to 7. This is because the domain of the function is restricted to $x \leq 7$, and the function is only defined for values of $x$ within this restricted domain.<br/> **Frequently Asked Questions (FAQs) about the Square Root Function**$

The square root function is a fundamental concept in mathematics that has numerous applications in various fields. However, it can be a bit tricky to understand, especially when it comes to its domain and restrictions. In this article, we will answer some frequently asked questions about the square root function to help you better understand this concept.

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

A: The domain of the square root function is all non-negative real numbers, i.e., $x \geq 0$ . This means that the function is only defined for values of $x$ that are greater than or equal to 0.

Q: What happens if I input a negative number into the square root function?

A: If you input a negative number into the square root function, the function will return an imaginary or complex number. This is because the square root of a negative number is not a real number.

Q: Can I restrict the domain of the square root function?

A: Yes, you can restrict the domain of the square root function. For example, if you want to restrict the domain to $x \leq 7$ , the function will only be defined for values of $x$ less than or equal to 7.

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

A: To determine the domain of a square root function with a restriction, you need to consider the restriction and the domain of the function. For example, if the function is $f(x) = \sqrt{x}$ and the domain is restricted to $x \leq 7$ , the function will only be defined for values of $x$ less than or equal to 7.

Q: Can I add or subtract a number from the $x$ -term inside the radical?

A: Yes, you can add or subtract a number from the $x$ -term inside the radical. For example, if you have the function $f(x) = \sqrt{x}$ and you want to add 7 to the $x$ -term, the function becomes $f(x) = \sqrt{x + 7}$ . However, you need to consider the domain of the function and the restriction.

Q: How do I determine the correct statement when the domain of the square root function is restricted?

A: To determine the correct statement when the domain of the square root function is restricted, you need to consider the restriction and the domain of the function. For example, if the function is $f(x) = \sqrt{x}$ and the domain is restricted to $x \leq 7$ , the correct statement is that the function is defined for values of $x$ less than or equal to 7.

Q: Can I multiply the radical by a negative number?

A: Yes, you can multiply the radical by a negative number. For example, if you have the function $f(x) = \sqrt{x}$ and you want to multiply the radical by -1, the function becomes $f(x) = -\sqrt{x}$ . However, you need to consider the domain of the function and the restriction.

Q: How do I evaluate the square root function with a restriction?

A: To evaluate the square root function with a restriction, you need to consider the restriction and the domain of the function. For example, if the function is $f(x) = \sqrt{x}$ and the domain is restricted to $x \leq 7$ , you need to input a value of $x$ that is less than or equal to 7.

Conclusion

In conclusion, the square root function is a fundamental concept in mathematics that has numerous applications in various fields. However, it can be a bit tricky to understand, especially when it comes to its domain and restrictions. By answering some frequently asked questions about the square root function, we hope to have helped you better understand this concept and how to work with it.