If The Domain Of The Square Root Function F ( X F(x F ( X ] Is X ≤ 7 X \leq 7 X ≤ 7 , Which Statement Must Be True?A. 7 Is Subtracted From The X X X -term Inside The Radical.B. The Radical Is Multiplied By A Negative Number.C. 7 Is Added To The

The square root function, denoted as $f(x) = \sqrt{x}$, is a fundamental concept in mathematics that deals with finding the value of a number that, when multiplied by itself, gives the original number. However, the domain of the square root function is restricted to non-negative real numbers, i.e., $x \geq 0$. This is because the square of any real number is always non-negative.

Restricting the Domain of the Square Root Function

In this problem, we are given that the domain of the square root function $f(x)$ is $x \leq 7$. This means that the function is defined for all real numbers less than or equal to 7. To understand which statement must be true, we need to analyze the given options.

Option 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, i.e., the function is defined for all real numbers greater than or equal to 0. However, this contradicts the given domain of the function, which is $x \leq 7$. Therefore, option A is not true.

Option 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 defined for all real numbers, but the output is negated. However, this does not affect the domain of the function, which is still $x \leq 7$. Therefore, option B is not true.

Option 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, i.e., the function is defined for all real numbers less than or equal to 7. This is consistent with the given domain of the function, which is $x \leq 7$. Therefore, option C is true.

Conclusion

In conclusion, the statement that must be true is that 7 is added to the $x$-term inside the radical. This is because the domain of the square root function is restricted to non-negative real numbers, and adding 7 to the $x$-term inside the radical shifts the domain to the left by 7 units, making it consistent with the given domain of the function.

Understanding the Concept of Domain

The concept of domain is crucial in mathematics, especially when dealing with functions. 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 restricted to non-negative real numbers. However, when the domain is restricted to $x \leq 7$, we need to analyze the given options to determine which statement must be true.

Analyzing the Options

When analyzing the options, we need to consider the effect of each option on the domain of the function. Option A subtracts 7 from the $x$-term inside the radical, which shifts the domain to the right by 7 units. Option B multiplies the radical by a negative number, which negates the output but does not affect the domain. Option C adds 7 to the $x$-term inside the radical, which shifts the domain to the left by 7 units.

Conclusion

In conclusion, the statement that must be true is that 7 is added to the $x$-term inside the radical. This is because the domain of the square root function is restricted to non-negative real numbers, and adding 7 to the $x$-term inside the radical shifts the domain to the left by 7 units, making it consistent with the given domain of the function.

Final Thoughts

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

A: The domain of the square root function is the set of all non-negative real numbers, i.e., $x \geq 0$. This is because the square of any real number is always non-negative.

Q: What happens if the domain of the square root function is restricted to $x \leq 7$?

A: If the domain of the square root function is restricted to $x \leq 7$, the function is defined for all real numbers less than or equal to 7. This means that the function is shifted to the left by 7 units.

Q: Which statement must be true if the domain of the square root function is restricted to $x \leq 7$?

A: The statement that must be true is that 7 is added to the $x$-term inside the radical. This is because adding 7 to the $x$-term inside the radical shifts the domain to the left by 7 units, making it consistent with the given domain of the function.

Q: What is the effect of subtracting 7 from the $x$-term inside the radical?

A: Subtracting 7 from the $x$-term inside the radical shifts the domain to the right by 7 units. This means that the function is defined for all real numbers greater than or equal to 0, which contradicts the given domain of the function.

Q: What is the effect of multiplying the radical by a negative number?

A: Multiplying the radical by a negative number negates the output but does not affect the domain. This means that the function is still defined for all real numbers, but the output is negated.

Q: Can the domain of the square root function be restricted to any value?

A: No, the domain of the square root function cannot be restricted to any value. The domain is always restricted to non-negative real numbers, i.e., $x \geq 0$.

Q: How can we determine the correct statement if the domain of the square root function is restricted?

A: To determine the correct statement, we need to analyze the effect of each option on the domain of the function. We need to consider whether the option shifts the domain to the left or right, and whether it affects the output.

Q: What is the importance of understanding the domain of a function?

A: Understanding the domain of a function is crucial in mathematics, especially when dealing with functions. The domain of a function is the set of all possible input values for which the function is defined. If the domain is not correctly understood, it can lead to incorrect results and conclusions.

Q: Can the domain of a function be changed?

A: Yes, the domain of a function can be changed. However, the new domain must be consistent with the definition of the function. For example, if the domain of the square root function is restricted to $x \leq 7$, the function is still defined for all real numbers less than or equal to 7.

Q: How can we find the correct statement if the domain of the square root function is restricted?

A: To find the correct statement, we need to analyze the effect of each option on the domain of the function. We need to consider whether the option shifts the domain to the left or right, and whether it affects the output. We can use the following steps:

1. Analyze the given options.
2. Determine the effect of each option on the domain of the function.
3. Choose the option that is consistent with the given domain of the function.

By following these steps, we can determine the correct statement if the domain of the square root function is restricted.