Which Function Has A Domain Of $x \geq 5$ And A Range Of $y \leq 3$?A. $y = \sqrt{x-5} + 3$ B. $y = \sqrt{x+5} - 3$ C. $y = -\sqrt{x-5} + 3$ D. $y = -\sqrt{x+5} - 3$

In mathematics, functions are used to describe relationships between variables. A function's domain is the set of all possible input values, while its range is the set of all possible output values. Understanding the domain and range of a function is crucial in various mathematical and real-world applications.

Domain and Range Basics

The domain of a function is the set of all possible input values, or x-values, that can be plugged into the function. It is usually denoted by the symbol "D" or "domain." On the other hand, the range of a function is the set of all possible output values, or y-values, that the function can produce. It is usually denoted by the symbol "R" or "range."

Domain of $x \geq 5$

The given domain is $x \geq 5$. This means that the input value, x, must be greater than or equal to 5. In other words, the function can only accept values of x that are 5 or greater.

Range of $y \leq 3$

The given range is $y \leq 3$. This means that the output value, y, must be less than or equal to 3. In other words, the function can only produce values of y that are 3 or less.

Analyzing the Options

Now, let's analyze each of the given options to determine which one has a domain of $x \geq 5$ and a range of $y \leq 3$.

Option A: $y = \sqrt{x-5} + 3$

For this option, the expression inside the square root must be non-negative, since the square root of a negative number is undefined. Therefore, we must have $x-5 \geq 0$, which implies $x \geq 5$. This matches the given domain.

However, the range of this function is not limited to $y \leq 3$. As x increases, the value of y will also increase, since the square root function is increasing. Therefore, this option does not match the given range.

Option B: $y = \sqrt{x+5} - 3$

For this option, the expression inside the square root must be non-negative, since the square root of a negative number is undefined. Therefore, we must have $x+5 \geq 0$, which implies $x \geq -5$. This does not match the given domain.

Option C: $y = -\sqrt{x-5} + 3$

For this option, the expression inside the square root must be non-negative, since the square root of a negative number is undefined. Therefore, we must have $x-5 \geq 0$, which implies $x \geq 5$. This matches the given domain.

As x increases, the value of y will decrease, since the square root function is increasing and the negative sign is present. Therefore, this option matches the given range.

Option D: $y = -\sqrt{x+5} - 3$

For this option, the expression inside the square root must be non-negative, since the square root of a negative number is undefined. Therefore, we must have $x+5 \geq 0$, which implies $x \geq -5$. This does not match the given domain.

Conclusion

Based on the analysis above, the correct option is C. $y = -\sqrt{x-5} + 3$. This option has a domain of $x \geq 5$ and a range of $y \leq 3$, as required.

Key Takeaways

• The domain of a function is the set of all possible input values.
• The range of a function is the set of all possible output values.
• Understanding the domain and range of a function is crucial in various mathematical and real-world applications.
• The given domain and range can be used to determine which option is correct.

Final Answer

In the previous article, we discussed the concept of domain and range in functions. We also analyzed a specific function to determine which option had a domain of $x \geq 5$ and a range of $y \leq 3$. In this article, we will provide a Q&A section to help clarify any doubts and provide more information on this topic.

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values, or x-values, that can be plugged into the function.

Q: What is the range of a function?

A: The range of a function is the set of all possible output values, or y-values, that the function can produce.

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 has a square root, the expression inside the square root must be non-negative. If the function has a fraction, the denominator cannot be zero.

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

A: To determine the range of a function, you need to consider the possible output values. For example, if the function is a linear function, the range will be all real numbers. If the function is a quadratic function, the range will be all real numbers except for the vertex.

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

A: The domain is the set of all possible input values, while the range is the set of all possible 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 $f(x) = x^2$ has a domain of all real numbers.

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

A: Yes, a function can have a range of all real numbers. For example, the function $f(x) = x$ has a range of all real numbers.

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

A: To graph a function with a given domain and range, you need to consider the following steps:

1. Determine the domain and range of the function.
2. Plot the function on a coordinate plane.
3. Use the domain and range to determine the x and y intercepts.
4. Use the domain and range to determine the asymptotes.

Q: What is the importance of understanding domain and range?

A: Understanding domain and range is crucial in various mathematical and real-world applications. It helps you to:

1. Determine the possible input and output values of a function.
2. Graph a function on a coordinate plane.
3. Analyze the behavior of a function.
4. Solve equations and inequalities.

Conclusion

In this article, we provided a Q&A section to help clarify any doubts and provide more information on the concept of domain and range in functions. We hope that this article has been helpful in understanding this important topic.

Key Takeaways

• The domain of a function is the set of all possible input values.
• The range of a function is the set of all possible output values.
• Understanding the domain and range of a function is crucial in various mathematical and real-world applications.
• The given domain and range can be used to determine which option is correct.

Final Answer

The final answer is C.