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

by ADMIN 209 views

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'll explore the domain and range of the function f(x)=βˆ’2xβˆ’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)=βˆ’2xβˆ’7+1f(x) = -2 \sqrt{x-7} + 1, we need to consider the expression inside the square root. The expression xβˆ’7x-7 must be non-negative, as the square root of a negative number is undefined in the real number system.

x - 7 \geq 0

Solving this inequality, we get:

x \geq 7

This means that the domain of the function f(x)=βˆ’2xβˆ’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)=βˆ’2xβˆ’7+1f(x) = -2 \sqrt{x-7} + 1, we need to consider the behavior of the function as xx approaches positive and negative infinity.

As xx approaches positive infinity, the value of xβˆ’7\sqrt{x-7} approaches infinity, and the value of βˆ’2xβˆ’7-2 \sqrt{x-7} approaches negative infinity. Therefore, the range of the function f(x)=βˆ’2xβˆ’7+1f(x) = -2 \sqrt{x-7} + 1 is all real numbers less than or equal to 1.

Analyzing the Statements

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

A. -6 is in the domain of f(x)f(x) but not in the range of f(x)f(x).

B. -6 is not in the domain of f(x)f(x) but is in the range of f(x)f(x).

C. -6 is in the domain of f(x)f(x) and is in the range of f(x)f(x).

From our analysis, we know that the domain of the function f(x)=βˆ’2xβˆ’7+1f(x) = -2 \sqrt{x-7} + 1 is all real numbers greater than or equal to 7. Therefore, -6 is not in the domain of f(x)f(x).

We also know that the range of the function f(x)=βˆ’2xβˆ’7+1f(x) = -2 \sqrt{x-7} + 1 is all real numbers less than or equal to 1. Therefore, -6 is not in the range of f(x)f(x).

Conclusion

Based on our analysis, we can conclude that the correct statement is:

B. -6 is not in the domain of f(x)f(x) but is in the range of f(x)f(x).

This statement accurately describes the domain and range of the function f(x)=βˆ’2xβˆ’7+1f(x) = -2 \sqrt{x-7} + 1.

Final Thoughts

Understanding the domain and range of a function is crucial in mathematics. It helps us to determine the possible input and output values of a function, which is essential in solving problems and making predictions. In this article, we have explored the domain and range of the function f(x)=βˆ’2xβˆ’7+1f(x) = -2 \sqrt{x-7} + 1 and determined which statement best describes it. We hope that this article has provided valuable insights and knowledge on this topic.

References

  • [1] "Functions" by Khan Academy
  • [2] "Domain and Range" by Math Open Reference
  • [3] "Square Root Function" by Wolfram MathWorld

Further Reading

  • [1] "Introduction to Functions" by MIT OpenCourseWare
  • [2] "Domain and Range of a Function" by Purplemath
  • [3] "Square Root Function" by Mathway

Glossary

  • Domain: The set of all possible input values for which the function is defined.
  • Range: The set of all possible output values.
  • Square Root Function: A function that takes a non-negative input and returns its square root.
    Q&A: Domain and Range of a Function =====================================

In our previous article, we explored the domain and range of the function f(x)=βˆ’2xβˆ’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 less than or equal to 1. In this article, we'll 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's the set of all possible x-values that can be plugged into the function.

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 make sure that the input value is non-negative. If the function involves a fraction, you need to make sure that the denominator is not equal to zero.

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's the set of all possible y-values that can be produced by the function.

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

A: To determine the range of a function, you need to consider the behavior of the function as the input values approach positive and negative infinity. You also need to consider any restrictions on the output values.

Q: Can a function have a domain of all real numbers?

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

Q: Can a function have a range of all real numbers?

A: No, a function cannot have a range of all real numbers. For example, the function f(x)=xf(x) = \sqrt{x} has a range of all non-negative 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 and range that are the same?

A: Yes, a function can have a domain and range that are the same. For example, the function f(x)=xf(x) = x has a domain and range of all real numbers.

Q: How do I graph a function with a domain and range?

A: To graph a function with a domain and range, you need to plot the function on a coordinate plane. You also need to indicate the domain and range of the function on the graph.

References

  • [1] "Functions" by Khan Academy
  • [2] "Domain and Range" by Math Open Reference
  • [3] "Square Root Function" by Wolfram MathWorld

Further Reading

  • [1] "Introduction to Functions" by MIT OpenCourseWare
  • [2] "Domain and Range of a Function" by Purplemath
  • [3] "Square Root Function" by Mathway

Glossary

  • Domain: The set of all possible input values for which the function is defined.
  • Range: The set of all possible output values.
  • One-to-One Function: A function that passes the horizontal line test.
  • Horizontal Line Test: A test used to determine if a function is one-to-one.
  • Coordinate Plane: A plane with x and y axes used to graph functions.