Which Statement Is True About The Function F ( X ) = X F(x)=\sqrt{x} F ( X ) = X ​ ?A. The Domain Of The Graph Is All Real Numbers.B. The Range Of The Graph Is All Real Numbers.C. The Domain Of The Graph Is All Real Numbers Less Than Or Equal To 0.D. The Range Of The

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Introduction to the Function

The function f(x)=xf(x)=\sqrt{x} is a fundamental concept in mathematics, particularly in algebra and calculus. It is a type of function known as a root function, which returns the square root of a given input. In this article, we will explore the properties of the function f(x)=xf(x)=\sqrt{x}, specifically its domain and range.

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 function f(x)=xf(x)=\sqrt{x}, the domain is all real numbers greater than or equal to 0. This is because the square root of a negative number is not a real number, and therefore, the function is not defined for negative values of x.

The domain of the function f(x)=xf(x)=\sqrt{x} is all real numbers greater than or equal to 0.

Range of the Function

The range of a function is the set of all possible output values for which the function is defined. In the case of the function f(x)=xf(x)=\sqrt{x}, the range is all real numbers greater than or equal to 0. This is because the square root of any non-negative real number is also a non-negative real number.

The range of the function f(x)=xf(x)=\sqrt{x} is all real numbers greater than or equal to 0.

Analyzing the Options

Now that we have a clear understanding of the domain and range of the function f(x)=xf(x)=\sqrt{x}, let's analyze the options provided:

A. The domain of the graph is all real numbers. B. The range of the graph is all real numbers. C. The domain of the graph is all real numbers less than or equal to 0. D. The range of the graph is all real numbers greater than or equal to 0.

Conclusion

Based on our analysis, we can conclude that the correct statement about the function f(x)=xf(x)=\sqrt{x} is:

The domain of the graph is all real numbers greater than or equal to 0, and the range of the graph is all real numbers greater than or equal to 0.

This statement is supported by the fact that the square root of a negative number is not a real number, and therefore, the function is not defined for negative values of x. Additionally, the square root of any non-negative real number is also a non-negative real number, which means that the range of the function is all real numbers greater than or equal to 0.

Final Thoughts

In conclusion, the function f(x)=xf(x)=\sqrt{x} is a fundamental concept in mathematics that has a specific domain and range. Understanding the properties of this function is essential for solving mathematical problems and making informed decisions in various fields. By analyzing the options provided, we can conclude that the correct statement about the function f(x)=xf(x)=\sqrt{x} is that the domain of the graph is all real numbers greater than or equal to 0, and the range of the graph is all real numbers greater than or equal to 0.

Key Takeaways

  • The domain of the function f(x)=xf(x)=\sqrt{x} is all real numbers greater than or equal to 0.
  • The range of the function f(x)=xf(x)=\sqrt{x} is all real numbers greater than or equal to 0.
  • The square root of a negative number is not a real number.
  • The square root of any non-negative real number is also a non-negative real number.

Recommendations

  • For further study, explore the properties of other root functions, such as the cube root function.
  • Practice solving mathematical problems that involve the function f(x)=xf(x)=\sqrt{x}.
  • Apply the concepts learned in this article to real-world problems in various fields, such as physics, engineering, and economics.

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 real numbers greater than or equal to 0. This is because the square root of a negative number is not a real number, and therefore, the function is not defined for negative values of x.

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 real numbers greater than or equal to 0. This is because the square root of any non-negative real number is also a non-negative real number.

Q: Can the function f(x)=xf(x)=\sqrt{x} be defined for negative values of x?

A: No, the function f(x)=xf(x)=\sqrt{x} cannot be defined for negative values of x. This is because the square root of a negative number is not a real number.

Q: What happens if we try to take the square root of a negative number?

A: If we try to take the square root of a negative number, we will get an imaginary number. For example, −1\sqrt{-1} is equal to ii, where ii is the imaginary unit.

Q: Can we use the function f(x)=xf(x)=\sqrt{x} to find the square root of a negative number?

A: No, we cannot use the function f(x)=xf(x)=\sqrt{x} to find the square root of a negative number. This is because the function is only defined for non-negative real numbers.

Q: How do we find the square root of a negative number?

A: To find the square root of a negative number, we can use the imaginary unit ii. For example, −1\sqrt{-1} is equal to ii.

Q: Can we use the function f(x)=xf(x)=\sqrt{x} to solve equations involving square roots?

A: Yes, we can use the function f(x)=xf(x)=\sqrt{x} to solve equations involving square roots. For example, we can use the function to solve equations of the form x2=ax^2 = a, where aa is a positive real number.

Q: How do we use the function f(x)=xf(x)=\sqrt{x} to solve equations involving square roots?

A: To use the function f(x)=xf(x)=\sqrt{x} to solve equations involving square roots, we can take the square root of both sides of the equation. For example, if we have the equation x2=4x^2 = 4, we can take the square root of both sides to get x=±2x = \pm 2.

Q: Can we use the function f(x)=xf(x)=\sqrt{x} to solve inequalities involving square roots?

A: Yes, we can use the function f(x)=xf(x)=\sqrt{x} to solve inequalities involving square roots. For example, we can use the function to solve inequalities of the form x2>ax^2 > a, where aa is a positive real number.

Q: How do we use the function f(x)=xf(x)=\sqrt{x} to solve inequalities involving square roots?

A: To use the function f(x)=xf(x)=\sqrt{x} to solve inequalities involving square roots, we can take the square root of both sides of the inequality. For example, if we have the inequality x2>4x^2 > 4, we can take the square root of both sides to get x>±2x > \pm 2.

Q: Can we use the function f(x)=xf(x)=\sqrt{x} to solve systems of equations involving square roots?

A: Yes, we can use the function f(x)=xf(x)=\sqrt{x} to solve systems of equations involving square roots. For example, we can use the function to solve systems of equations of the form x2+y2=ax^2 + y^2 = a and x+y=bx + y = b, where aa and bb are positive real numbers.

Q: How do we use the function f(x)=xf(x)=\sqrt{x} to solve systems of equations involving square roots?

A: To use the function f(x)=xf(x)=\sqrt{x} to solve systems of equations involving square roots, we can substitute the expression for yy from the second equation into the first equation. For example, if we have the system of equations x2+y2=4x^2 + y^2 = 4 and x+y=2x + y = 2, we can substitute y=2−xy = 2 - x into the first equation to get x2+(2−x)2=4x^2 + (2 - x)^2 = 4.

Q: Can we use the function f(x)=xf(x)=\sqrt{x} to solve optimization problems involving square roots?

A: Yes, we can use the function f(x)=xf(x)=\sqrt{x} to solve optimization problems involving square roots. For example, we can use the function to solve optimization problems of the form max⁡x2\max x^2 subject to x≥0x \geq 0.

Q: How do we use the function f(x)=xf(x)=\sqrt{x} to solve optimization problems involving square roots?

A: To use the function f(x)=xf(x)=\sqrt{x} to solve optimization problems involving square roots, we can use the method of Lagrange multipliers. For example, if we have the optimization problem max⁡x2\max x^2 subject to x≥0x \geq 0, we can use the method of Lagrange multipliers to find the maximum value of x2x^2 subject to the constraint x≥0x \geq 0.

Key Takeaways

  • The domain of the function f(x)=xf(x)=\sqrt{x} is all real numbers greater than or equal to 0.
  • The range of the function f(x)=xf(x)=\sqrt{x} is all real numbers greater than or equal to 0.
  • The square root of a negative number is not a real number.
  • The square root of any non-negative real number is also a non-negative real number.
  • We can use the function f(x)=xf(x)=\sqrt{x} to solve equations involving square roots.
  • We can use the function f(x)=xf(x)=\sqrt{x} to solve inequalities involving square roots.
  • We can use the function f(x)=xf(x)=\sqrt{x} to solve systems of equations involving square roots.
  • We can use the function f(x)=xf(x)=\sqrt{x} to solve optimization problems involving square roots.

Recommendations

  • For further study, explore the properties of other root functions, such as the cube root function.
  • Practice solving mathematical problems that involve the function f(x)=xf(x)=\sqrt{x}.
  • Apply the concepts learned in this article to real-world problems in various fields, such as physics, engineering, and economics.