Which Statement Best Describes $f(x)=-2 \sqrt{x-7}+1$?A. -6 Is In The Domain Of $f(x$\] But Not In The Range Of $f(x$\].B. -6 Is Not In The Domain Of $f(x$\] But Is In The Range Of $f(x$\].C. -6 Is In The Domain

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Introduction

When dealing with functions, it's essential to understand the concepts 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 will explore the domain and range of the function f(x)=−2x−7+1f(x)=-2 \sqrt{x-7}+1 and determine which statement best describes it.

Domain of a 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)=−2x−7+1f(x)=-2 \sqrt{x-7}+1, we need to consider the restrictions on the input values.

  • Square Root Function: The square root function x\sqrt{x} is defined only for non-negative real numbers. Therefore, the expression inside the square root must be non-negative, i.e., x−7≥0x-7 \geq 0.
  • Domain Restriction: Solving the inequality x−7≥0x-7 \geq 0, we get x≥7x \geq 7. This means that the domain of the function f(x)=−2x−7+1f(x)=-2 \sqrt{x-7}+1 is all real numbers greater than or equal to 7.

Range of a Function

The range of a function is the set of all possible output values. To determine the range of the function f(x)=−2x−7+1f(x)=-2 \sqrt{x-7}+1, we need to consider the possible values of the output.

  • Square Root Function: The square root function x\sqrt{x} is defined only for non-negative real numbers. Therefore, the expression inside the square root must be non-negative, i.e., x−7≥0x-7 \geq 0.
  • Range Restriction: Since the square root function is defined only for non-negative real numbers, the output of the function f(x)=−2x−7+1f(x)=-2 \sqrt{x-7}+1 will always be non-negative. Additionally, the function has a minimum value of 1, which occurs when x=7x=7. Therefore, the range of the function f(x)=−2x−7+1f(x)=-2 \sqrt{x-7}+1 is all real numbers greater than or equal to 1.

Analyzing the Statements

Now that we have determined the domain and range of the function f(x)=−2x−7+1f(x)=-2 \sqrt{x-7}+1, let's analyze the statements:

  • Statement A: -6 is in the domain of f(x)f(x) but not in the range of f(x)f(x). This statement is incorrect because -6 is not in the domain of f(x)f(x), as it is less than 7.
  • Statement B: -6 is not in the domain of f(x)f(x) but is in the range of f(x)f(x). This statement is incorrect because -6 is not in the range of f(x)f(x), as the range of f(x)f(x) is all real numbers greater than or equal to 1.
  • Statement C: -6 is in the domain of f(x)f(x) but not in the range of f(x)f(x). This statement is incorrect because -6 is not in the domain of f(x)f(x), as it is less than 7.

Conclusion

In conclusion, the correct statement is not among the options provided. The domain of the function f(x)=−2x−7+1f(x)=-2 \sqrt{x-7}+1 is all real numbers greater than or equal to 7, and the range is all real numbers greater than or equal to 1. Therefore, none of the statements accurately describe the function.

Final Answer

Introduction

In our previous article, we explored the domain and range of the function f(x)=−2x−7+1f(x)=-2 \sqrt{x-7}+1. We determined that the domain of the function is all real numbers greater than or equal to 7, and the range is all real numbers greater than or equal to 1. In this article, we will answer some frequently asked questions about the domain and range of a function.

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible x-values for which the function is defined.

Q: How do I determine the domain of a function?

A: To determine the domain of a function, you need to consider any restrictions on the input values. For example, if the function involves a square root, you need to ensure that the expression inside the square root is non-negative. You also need to consider any other restrictions, such as division by zero or taking the logarithm of a non-positive number.

Q: What is the range of a function?

A: The range of a function is the set of all possible output values. In other words, it is the set of all possible y-values for which the function is defined.

Q: How do I determine the range of a function?

A: To determine the range of a function, you need to consider the possible values of the output. You can do this by analyzing the function and determining the minimum and maximum values of the output.

Q: Can a function have a domain that is all real numbers?

A: Yes, a function can have a domain that is all real numbers. For example, the function f(x)=x2f(x)=x^2 has a domain that is all real numbers.

Q: Can a function have a range that is all real numbers?

A: No, a function cannot have a range that is all real numbers. The range of a function is always a subset of the real numbers.

Q: How do I determine if a function is one-to-one?

A: To determine if a function is one-to-one, you need to check if the function passes the horizontal line test. If the function passes the horizontal line test, then it is one-to-one.

Q: What is the difference between the domain and range of a function?

A: The domain of a function is the set of all possible input values, while the range of a function is the set of all possible output values.

Q: Can a function have a domain that is a subset of the real numbers?

A: Yes, a function can have a domain that is a subset of the real numbers. For example, the function f(x)=xf(x)=\sqrt{x} has a domain that is all non-negative real numbers.

Q: Can a function have a range that is a subset of the real numbers?

A: Yes, a function can have a range that is a subset of the real numbers. For example, the function f(x)=sin⁡(x)f(x)=\sin(x) has a range that is the set of all real numbers between -1 and 1.

Conclusion

In conclusion, the domain and range of a function are essential concepts in mathematics. Understanding the domain and range of a function can help you determine the possible input and output values of the function. We hope that this article has helped you understand the domain and range of a function better.

Final Answer

The final answer is that the domain and range of a function are essential concepts in mathematics, and understanding them can help you determine the possible input and output values of the function.