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, as the square of any negative number is positive. In this article, we will explore the implications of the domain of the square root function being $x \leq 7$ and determine which statement must be true.

The Domain of the Square Root 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 domain is restricted to non-negative real numbers, which can be represented as $x \geq 0$. However, in this problem, the domain is specified as $x \leq 7$, which means that the square root function is defined for all real numbers less than or equal to 7.

Analyzing the Statements

Now, let's analyze the three statements given in the problem and determine which one must be true.

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 would be $x - 7 \geq 0$, which simplifies to $x \geq 7$. However, this contradicts the given domain of $x \leq 7$. 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 domain of the function would be the same as the original function, which is $x \geq 0$. However, this contradicts the given domain of $x \leq 7$. 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 would be $x + 7 \geq 0$, which simplifies to $x \geq -7$. However, this does not contradict the given domain of $x \leq 7$. In fact, this statement is consistent with the given domain, as the function is defined for all real numbers less than or equal to 7.

Conclusion

In conclusion, the statement that must be true is C. 7 is added to the $x$-term inside the radical. This is because the function $f(x) = \sqrt{x + 7}$ is defined for all real numbers less than or equal to 7, which is consistent with the given domain.

Understanding the Implications

The implications of the domain of the square root function being $x \leq 7$ are that the function is defined for all real numbers less than or equal to 7. This means that the function can take on any value between 0 and 7, inclusive. The function is not defined for any real number greater than 7, as the square root of a negative number is not a real number.

Real-World Applications

The square root function has many real-world applications, including:

• Physics: The square root function is used to calculate the speed of an object, given its kinetic energy.
• Engineering: The square root function is used to calculate the stress on a material, given its cross-sectional area and the force applied to it.
• Finance: The square root function is used to calculate the volatility of a stock, given its historical price data.

Conclusion

In conclusion, the domain of the square root function being $x \leq 7$ has significant implications for the function's behavior and applications. The statement that must be true is C. 7 is added to the $x$-term inside the radical. The square root function has many real-world applications, including physics, engineering, and finance.

References

• Khan Academy: Square Root Function
• Math Is Fun: Square Root Function
• Wolfram MathWorld: Square Root Function
  Frequently Asked Questions (FAQs) about the Square Root Function ====================================================================

The square root function is a fundamental concept in mathematics that deals with finding the value of a number that, when multiplied by itself, gives the original number. In this article, we will answer some frequently asked questions about the square root function.

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

A: The domain of the square root function is all non-negative real numbers, which can be represented as $x \geq 0$. However, in this problem, the domain is specified as $x \leq 7$, which means that the square root function is defined for all real numbers less than or equal to 7.

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

A: The range of the square root function is all non-negative real numbers, which can be represented as $y \geq 0$. This means that the output of the function will always be a non-negative real number.

Q: How do I calculate the square root of a number?

A: To calculate the square root of a number, you can use a calculator or a computer program. Alternatively, you can use the following formula:

$$\sqrt{x} = \pm \sqrt{\left| x \right|}$$

This formula states that the square root of a number is equal to the positive or negative square root of the absolute value of the number.

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

A: The square root and the cube root are both roots of a number, but they are different. The square root of a number is a value that, when multiplied by itself, gives the original number. The cube root of a number is a value that, when multiplied by itself three times, gives the original number.

Q: Can I take the square root of a negative number?

A: No, you cannot take the square root of a negative number. The square root of a negative number is not a real number, as the square of any real number is always non-negative.

Q: What is the square root of 0?

A: The square root of 0 is 0. This is because 0 multiplied by 0 gives 0.

Q: What is the square root of a fraction?

A: The square root of a fraction is a value that, when multiplied by itself, gives the original fraction. For example, the square root of 1/4 is 1/2, as 1/2 multiplied by 1/2 gives 1/4.

Q: Can I take the square root of a decimal number?

A: Yes, you can take the square root of a decimal number. For example, the square root of 0.25 is 0.5, as 0.5 multiplied by 0.5 gives 0.25.

Q: What is the square root of a complex number?

A: The square root of a complex number is a value that, when multiplied by itself, gives the original complex number. For example, the square root of 3 + 4i is 1 + 2i, as (1 + 2i) multiplied by (1 + 2i) gives 3 + 4i.

Conclusion

In conclusion, the square root function is a fundamental concept in mathematics that deals with finding the value of a number that, when multiplied by itself, gives the original number. We have answered some frequently asked questions about the square root function, including its domain, range, and how to calculate it.

References

• Khan Academy: Square Root Function
• Math Is Fun: Square Root Function
• Wolfram MathWorld: Square Root Function