Which Function Has A Range That Includes -4?A. Y = X − 5 Y = \sqrt{x} - 5 Y = X ​ − 5 B. Y = X + 5 Y = \sqrt{x} + 5 Y = X ​ + 5 C. Y = X + 5 Y = \sqrt{x + 5} Y = X + 5 ​ D. Y = X − 5 Y = \sqrt{x - 5} Y = X − 5 ​

When dealing with functions, particularly those involving square roots, it's essential to understand the concept of range. The range of a function is the set of all possible output values it can produce for the given input values. In this article, we'll explore four different functions and determine which one has a range that includes -4.

Understanding the Basics of Square Root Functions

Before we dive into the specific functions, let's review the basics of square root functions. The square root function, denoted by $\sqrt{x}$, is defined as the inverse of the squaring function. It returns the value that, when squared, gives the original input value. For example, $\sqrt{16} = 4$ because $4^2 = 16$.

However, when dealing with square root functions, we need to consider the domain and range. The domain of a function is the set of all possible input values, while the range is the set of all possible output values. For the square root function, the domain is all non-negative real numbers, and the range is also all non-negative real numbers.

Analyzing the Given Functions

Now, let's analyze the four given functions:

A. $y = \sqrt{x} - 5$

This function involves subtracting 5 from the square root of x. To determine the range, we need to consider the possible values of x. Since the square root function has a domain of all non-negative real numbers, x can take any value greater than or equal to 0. However, when we subtract 5 from the square root of x, the minimum value of y will be -5, which is not included in the range of the square root function.

B. $y = \sqrt{x} + 5$

This function involves adding 5 to the square root of x. Similar to the previous function, the domain of x is all non-negative real numbers. However, when we add 5 to the square root of x, the minimum value of y will be 5, which is included in the range of the square root function.

C. $y = \sqrt{x + 5}$

This function involves adding 5 to x before taking the square root. The domain of x is all real numbers, but since we're adding 5 to x, the effective domain becomes all real numbers greater than or equal to -5. When we take the square root of x + 5, the minimum value of y will be 0, which is included in the range of the square root function.

D. $y = \sqrt{x - 5}$

This function involves subtracting 5 from x before taking the square root. The domain of x is all real numbers greater than or equal to 5, since we're subtracting 5 from x. When we take the square root of x - 5, the minimum value of y will be 0, which is included in the range of the square root function.

Conclusion

Based on our analysis, we can conclude that functions C and D have a range that includes -4. However, we need to be careful when considering the domain of each function. Function C has a domain of all real numbers greater than or equal to -5, while function D has a domain of all real numbers greater than or equal to 5.

Which Function Has a Range That Includes -4?

After careful analysis, we can conclude that function C, $y = \sqrt{x + 5}$, has a range that includes -4. This is because the domain of x is all real numbers, and when we add 5 to x, the effective domain becomes all real numbers greater than or equal to -5. When we take the square root of x + 5, the minimum value of y will be 0, which is included in the range of the square root function.

Final Answer

In our previous article, we explored the concept of range in square root functions and determined which function has a range that includes -4. In this article, we'll answer some frequently asked questions related to the range of square root functions.

Q: What is the range of a square root function?

A: The range of a square root function is all non-negative real numbers. This means that the output of the function will always be a non-negative value.

Q: Can the range of a square root function include negative numbers?

A: No, the range of a square root function cannot include negative numbers. This is because the square root of a negative number is undefined in the real number system.

Q: How does the domain of a square root function affect its range?

A: The domain of a square root function affects its range by determining the possible input values. If the domain is restricted to non-negative real numbers, the range will also be restricted to non-negative real numbers.

Q: Can the range of a square root function be restricted to a specific interval?

A: Yes, the range of a square root function can be restricted to a specific interval. For example, if we have a function $y = \sqrt{x - 5}$, the range will be restricted to non-negative real numbers greater than or equal to 5.

Q: How do we determine the range of a square root function with a restricted domain?

A: To determine the range of a square root function with a restricted domain, we need to consider the minimum value of the input values. If the minimum value is non-negative, the range will be all non-negative real numbers. If the minimum value is negative, the range will be restricted to non-negative real numbers greater than or equal to the absolute value of the minimum value.

Q: Can we use the range of a square root function to solve equations?

A: Yes, we can use the range of a square root function to solve equations. For example, if we have an equation $y = \sqrt{x - 5}$ and we know that $y = 3$, we can solve for x by squaring both sides of the equation and adding 5 to both sides.

Q: How do we graph a square root function with a restricted domain?

A: To graph a square root function with a restricted domain, we need to consider the domain and range of the function. We can use a graphing calculator or software to graph the function and visualize its behavior.

Q: Can we use the range of a square root function to model real-world phenomena?

A: Yes, we can use the range of a square root function to model real-world phenomena. For example, if we have a function that models the growth of a population, we can use the range of the function to determine the maximum population size.

Conclusion

In this article, we've answered some frequently asked questions related to the range of square root functions. We've discussed how the domain of a square root function affects its range, how to determine the range of a square root function with a restricted domain, and how to use the range of a square root function to solve equations and model real-world phenomena.

Final Tips

• Always consider the domain and range of a square root function when solving equations or modeling real-world phenomena.
• Use graphing calculators or software to visualize the behavior of a square root function with a restricted domain.
• Be careful when using the range of a square root function to solve equations, as the range may be restricted to a specific interval.

Common Mistakes to Avoid

• Assuming that the range of a square root function is all real numbers.
• Failing to consider the domain of a square root function when solving equations or modeling real-world phenomena.
• Using the range of a square root function to solve equations without considering the domain of the function.

Final Answer

The final answer is that the range of a square root function is all non-negative real numbers, and the domain of a square root function affects its range by determining the possible input values.