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

When dealing with functions, particularly those involving square roots, it's essential to understand the concept of domain and range. The domain of a function is the set of all possible input values for which the function is defined, while the range is the set of all possible output values. In this article, we'll delve into the function f(x)=xf(x)=\sqrt{x} and analyze its domain and range to determine which statement is true.

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}, we need to consider the values of xx that make the function defined. Since the square root of a negative number is not a real number, the function is only defined for non-negative values of xx. Therefore, 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. In the case of the function f(x)=xf(x)=\sqrt{x}, the output values are all non-negative real numbers. This is because the square root of any non-negative real number is always non-negative. Therefore, the range of the function f(x)=xf(x)=\sqrt{x} is all real numbers greater than or equal to 0.

Analyzing the Statements

Now that we've analyzed the domain and range of the function f(x)=xf(x)=\sqrt{x}, let's examine the given statements:

  • 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 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.

Therefore, the correct answer is:

  • C. The domain of the graph is all real numbers less than or equal to 0. (This is incorrect, the correct answer is all real numbers greater than or equal to 0)
  • D. The range of the graph is all real numbers greater than or equal to 0. (This is correct)

Final Answer

The final answer is D.

Introduction

In our previous article, we analyzed the domain and range of the function f(x)=xf(x)=\sqrt{x}. In this article, we'll address some frequently asked questions about this function to provide a deeper understanding of its properties.

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.

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 always non-negative.

Q: Is the function f(x)=xf(x)=\sqrt{x} defined for all real numbers?

A: No, the function f(x)=xf(x)=\sqrt{x} is not defined for all real numbers. It is only defined for non-negative real numbers.

Q: Is the function f(x)=xf(x)=\sqrt{x} a one-to-one function?

A: Yes, the function f(x)=xf(x)=\sqrt{x} is a one-to-one function. This means that each output value corresponds to exactly one input value.

Q: Is the function f(x)=xf(x)=\sqrt{x} an even function?

A: Yes, the function f(x)=xf(x)=\sqrt{x} is an even function. This means that f(−x)=f(x)f(-x) = f(x) for all xx in the domain of the function.

Q: Can the function f(x)=xf(x)=\sqrt{x} be inverted?

A: Yes, the function f(x)=xf(x)=\sqrt{x} can be inverted. The inverse function is f−1(x)=x2f^{-1}(x) = x^2.

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

A: The graph of the function f(x)=xf(x)=\sqrt{x} is a curve that starts at the origin (0,0) and increases without bound as xx increases. The curve is symmetric about the y-axis.

Conclusion

In this article, we've addressed some frequently asked questions about the function f(x)=xf(x)=\sqrt{x}. We've discussed its domain and range, its properties as a one-to-one and even function, and its invertibility. We've also described its graph and provided some examples of how to work with this function.

Final Answer

The final answer is that the function f(x)=xf(x)=\sqrt{x} is a well-defined function with a specific domain and range, and it has several interesting properties that make it useful in mathematics and other fields.

Additional Resources

  • For more information about the function f(x)=xf(x)=\sqrt{x}, see the article "Understanding the Function f(x)=xf(x)=\sqrt{x}: Domain and Range Analysis".
  • For more information about one-to-one and even functions, see the article "One-to-One and Even Functions: Properties and Examples".
  • For more information about invertible functions, see the article "Invertible Functions: Properties and Examples".