Which Statement Is True About The Function $f(x) = -\sqrt{x}$?A. It Has The Same Range But Not The Same Domain As The Function $f(x) = \sqrt{x}$.B. It Has The Same Range But Not The Same Domain As The Function $f(x)

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In mathematics, functions play a crucial role in describing the relationship between variables. A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). In this article, we will explore the function f(x)=−xf(x) = -\sqrt{x} and determine which statement is true about it.

The Function f(x)=−xf(x) = -\sqrt{x}

The function f(x)=−xf(x) = -\sqrt{x} is a square root function with a negative sign in front of it. This means that the function will take the square root of the input xx and then multiply it by −1-1. The domain of this function is all non-negative real numbers, i.e., x≥0x \geq 0. This is because the square root of a negative number is undefined in real numbers.

The Domain and Range of the Function

The domain of the function f(x)=−xf(x) = -\sqrt{x} is all non-negative real numbers, i.e., x≥0x \geq 0. This is because the square root of a negative number is undefined in real numbers. The range of the function is all non-positive real numbers, i.e., y≤0y \leq 0. This is because the negative sign in front of the square root function will always result in a non-positive output.

Comparing the Function with f(x)=xf(x) = \sqrt{x}

Now, let's compare the function f(x)=−xf(x) = -\sqrt{x} with the function f(x)=xf(x) = \sqrt{x}. The function f(x)=xf(x) = \sqrt{x} has the same domain as the function f(x)=−xf(x) = -\sqrt{x}, i.e., all non-negative real numbers. However, the range of the function f(x)=xf(x) = \sqrt{x} is all non-negative real numbers, i.e., y≥0y \geq 0. This is because the square root function will always result in a non-negative output.

Which Statement is True?

Now, let's determine which statement is true about the function f(x)=−xf(x) = -\sqrt{x}. Statement A says that the function has the same range but not the same domain as the function f(x)=xf(x) = \sqrt{x}. This statement is true because the function f(x)=−xf(x) = -\sqrt{x} has the same range as the function f(x)=xf(x) = \sqrt{x}, i.e., all non-positive real numbers, but it has a different domain, i.e., all non-negative real numbers.

Conclusion

In conclusion, the function f(x)=−xf(x) = -\sqrt{x} has the same range but not the same domain as the function f(x)=xf(x) = \sqrt{x}. This is because the function f(x)=−xf(x) = -\sqrt{x} has a different domain, i.e., all non-negative real numbers, but it has the same range, i.e., all non-positive real numbers.

Key Takeaways

  • The function f(x)=−xf(x) = -\sqrt{x} has the same range but not the same domain as the function f(x)=xf(x) = \sqrt{x}.
  • The domain of the function f(x)=−xf(x) = -\sqrt{x} is all non-negative real numbers, i.e., x≥0x \geq 0.
  • The range of the function f(x)=−xf(x) = -\sqrt{x} is all non-positive real numbers, i.e., y≤0y \leq 0.

Frequently Asked Questions

Q: What is the domain of the function f(x)=−xf(x) = -\sqrt{x}?

A: The domain of the function f(x)=−xf(x) = -\sqrt{x} is all non-negative real numbers, i.e., x≥0x \geq 0.

Q: What is the range of the function f(x)=−xf(x) = -\sqrt{x}?

A: The range of the function f(x)=−xf(x) = -\sqrt{x} is all non-positive real numbers, i.e., y≤0y \leq 0.

Q: Which statement is true about the function f(x)=−xf(x) = -\sqrt{x}?

A: Statement A is true, i.e., the function has the same range but not the same domain as the function f(x)=xf(x) = \sqrt{x}.

Q: How does the function f(x)=−xf(x) = -\sqrt{x} compare with the function f(x)=xf(x) = \sqrt{x}?

In our previous article, we explored the function f(x)=−xf(x) = -\sqrt{x} and determined that it has the same range but not the same domain as the function f(x)=xf(x) = \sqrt{x}. In this article, we will continue to answer more questions about the function f(x)=−xf(x) = -\sqrt{x}.

Q: What is the difference between the function f(x)=−xf(x) = -\sqrt{x} and the function f(x)=xf(x) = \sqrt{x}?

A: The main difference between the function f(x)=−xf(x) = -\sqrt{x} and the function f(x)=xf(x) = \sqrt{x} is the sign in front of the square root. The function f(x)=xf(x) = \sqrt{x} has a positive sign, while the function f(x)=−xf(x) = -\sqrt{x} has a negative sign. This means that the function f(x)=−xf(x) = -\sqrt{x} will always result in a non-positive output, while the function f(x)=xf(x) = \sqrt{x} will always result in a non-negative output.

Q: How does the function f(x)=−xf(x) = -\sqrt{x} behave for negative values of x?

A: The function f(x)=−xf(x) = -\sqrt{x} is not defined for negative values of x. This is because the square root of a negative number is undefined in real numbers. Therefore, the function f(x)=−xf(x) = -\sqrt{x} is only defined for non-negative values of x.

Q: Can the function f(x)=−xf(x) = -\sqrt{x} be used to model real-world phenomena?

A: Yes, the function f(x)=−xf(x) = -\sqrt{x} can be used to model real-world phenomena. For example, the function can be used to model the relationship between the distance of an object from a point and the time it takes for the object to reach that point. The function can also be used to model the relationship between the amount of a substance and the time it takes for the substance to decay.

Q: How does the function f(x)=−xf(x) = -\sqrt{x} compare with other functions?

A: The function f(x)=−xf(x) = -\sqrt{x} can be compared with other functions such as the function f(x)=x2f(x) = x^2 and the function f(x)=∣x∣f(x) = |x|. The function f(x)=x2f(x) = x^2 is a quadratic function that is always non-negative, while the function f(x)=∣x∣f(x) = |x| is an absolute value function that is always non-negative. The function f(x)=−xf(x) = -\sqrt{x} is different from these functions because it is always non-positive.

Q: Can the function f(x)=−xf(x) = -\sqrt{x} be used to solve equations?

A: Yes, the function f(x)=−xf(x) = -\sqrt{x} can be used to solve equations. For example, the equation f(x)=−x=0f(x) = -\sqrt{x} = 0 can be solved by setting x=0x = 0. This is because the function f(x)=−xf(x) = -\sqrt{x} is only defined for non-negative values of x, and when x is 0, the function is equal to 0.

Q: How does the function f(x)=−xf(x) = -\sqrt{x} relate to other mathematical concepts?

A: The function f(x)=−xf(x) = -\sqrt{x} relates to other mathematical concepts such as calculus and algebra. The function can be used to model the behavior of functions in calculus, and it can also be used to solve equations in algebra.

Q: Can the function f(x)=−xf(x) = -\sqrt{x} be used to model real-world phenomena in different fields?

A: Yes, the function f(x)=−xf(x) = -\sqrt{x} can be used to model real-world phenomena in different fields such as physics, engineering, and economics. For example, the function can be used to model the relationship between the distance of an object from a point and the time it takes for the object to reach that point in physics, or the relationship between the amount of a substance and the time it takes for the substance to decay in economics.

Key Takeaways

  • The function f(x)=−xf(x) = -\sqrt{x} has the same range but not the same domain as the function f(x)=xf(x) = \sqrt{x}.
  • The function f(x)=−xf(x) = -\sqrt{x} is not defined for negative values of x.
  • The function f(x)=−xf(x) = -\sqrt{x} can be used to model real-world phenomena in different fields.
  • The function f(x)=−xf(x) = -\sqrt{x} can be used to solve equations.

Frequently Asked Questions

Q: What is the difference between the function f(x)=−xf(x) = -\sqrt{x} and the function f(x)=xf(x) = \sqrt{x}?

A: The main difference between the function f(x)=−xf(x) = -\sqrt{x} and the function f(x)=xf(x) = \sqrt{x} is the sign in front of the square root.

Q: How does the function f(x)=−xf(x) = -\sqrt{x} behave for negative values of x?

A: The function f(x)=−xf(x) = -\sqrt{x} is not defined for negative values of x.

Q: Can the function f(x)=−xf(x) = -\sqrt{x} be used to model real-world phenomena?

A: Yes, the function f(x)=−xf(x) = -\sqrt{x} can be used to model real-world phenomena.

Q: How does the function f(x)=−xf(x) = -\sqrt{x} compare with other functions?

A: The function f(x)=−xf(x) = -\sqrt{x} can be compared with other functions such as the function f(x)=x2f(x) = x^2 and the function f(x)=∣x∣f(x) = |x|.

Q: Can the function f(x)=−xf(x) = -\sqrt{x} be used to solve equations?

A: Yes, the function f(x)=−xf(x) = -\sqrt{x} can be used to solve equations.