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

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 $f(x) = -2 \sqrt{x-7} + 1$ and determine which statement best describes it.

What is the Domain of a Function?

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 values of $x$ that can be plugged into the function without causing any problems. For the function $f(x) = -2 \sqrt{x-7} + 1$, we need to consider the expression inside the square root, which is $x-7$. Since the square root of a negative number is not defined in the real number system, we must ensure that $x-7$ is always non-negative.

Finding the Domain of $f(x)$

To find the domain of $f(x)$, we need to find the values of $x$ for which $x-7$ is non-negative. This can be expressed as the inequality $x-7 \geq 0$. Solving for $x$, we get $x \geq 7$. Therefore, the domain of $f(x)$ is all real numbers greater than or equal to 7.

What is the Range of a Function?

The range of a function is the set of all possible output values for which the function is defined. In other words, it's the set of all possible values of $f(x)$ that can be obtained by plugging in different values of $x$. For the function $f(x) = -2 \sqrt{x-7} + 1$, we need to consider the expression $-2 \sqrt{x-7} + 1$. Since the square root of a negative number is not defined in the real number system, we must ensure that $x-7$ is always non-negative.

Finding the Range of $f(x)$

To find the range of $f(x)$, we need to find the values of $f(x)$ for which $x-7$ is non-negative. Since $x-7$ is always non-negative for $x \geq 7$, we can plug in any value of $x$ greater than or equal to 7 into the function. This will give us a value of $f(x)$ that is always greater than or equal to $-2 \sqrt{0} + 1 = 1$. Therefore, the range of $f(x)$ is all real numbers greater than or equal to 1.

Which Statement Best Describes $f(x)$?

Now that we have found the domain and range of $f(x)$, we can determine which statement best describes it. Let's examine each statement:

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 of $f(x)$ and is in the range of $f(x)$.

Based on our analysis, we know that the domain of $f(x)$ is all real numbers greater than or equal to 7, and the range of $f(x)$ is all real numbers greater than or equal to 1. Therefore, statement A is the only one that accurately describes $f(x)$.

Conclusion

In conclusion, the domain of the function $f(x) = -2 \sqrt{x-7} + 1$ is all real numbers greater than or equal to 7, and the range of $f(x)$ is all real numbers greater than or equal to 1. Therefore, statement A is the only one that accurately describes $f(x)$. This demonstrates the importance of understanding the domain and range of a function in mathematics.

References

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

Additional Resources

• [1] "Domain and Range" by IXL
• [2] "Functions" by Purplemath
• [3] "Square Root" by Mathway
  Q&A: Domain and Range of Functions =====================================

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

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 values of $x$ that can be plugged into the function without causing any problems.

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

A: To find 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 can also use the following steps:

1. Identify any restrictions on the input values.
2. Express the restrictions as inequalities.
3. Solve the inequalities to find the domain of the function.

Q: What is the range of a function?

A: The range of a function is the set of all possible output values for which the function is defined. In other words, it's the set of all possible values of $f(x)$ that can be obtained by plugging in different values of $x$.

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

A: To find the range of a function, you need to consider the possible output values. You can use the following steps:

1. Identify the type of function (e.g., linear, quadratic, polynomial).
2. Determine the minimum and maximum values of the function.
3. Express the range as an interval or a set of 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 $f(x) = x$ has a domain and range of all real numbers.

Q: Can a function have a domain that is empty?

A: Yes, a function can have a domain that is empty. For example, the function $f(x) = \frac{1}{x}$ has a domain of all real numbers except 0.

Q: Can a function have a range that is empty?

A: Yes, a function can have a range that is empty. For example, the function $f(x) = \frac{1}{x}$ has a range of all real numbers except 0.

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

A: To determine if a function is one-to-one or onto, you need to consider the following:

• One-to-one: A function is one-to-one if each output value corresponds to exactly one input value.
• Onto: A function is onto if each output value is obtained by plugging in at least one input value.

You can use the following steps to determine if a function is one-to-one or onto:

1. Check if the function is injective (one-to-one).
2. Check if the function is surjective (onto).

Q: Can a function be both one-to-one and onto?

A: Yes, a function can be both one-to-one and onto. For example, the function $f(x) = x$ is both one-to-one and onto.

Conclusion

In conclusion, the domain and range of a function are essential concepts in mathematics. By understanding the domain and range of a function, you can determine its behavior and properties. We hope this Q&A article has helped you understand the domain and range of functions better.

References

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

Additional Resources

• [1] "Domain and Range" by IXL
• [2] "Functions" by Purplemath
• [3] "Square Root" by Mathway