What Are The Domain And Range Of The Function F ( X ) = − X + 3 − 2 F(x) = -\sqrt{x+3} - 2 F ( X ) = − X + 3 ​ − 2 ?A. Domain: − 3 ≤ X ≤ − 2 -3 \leq X \leq -2 − 3 ≤ X ≤ − 2 ; Range: Y ≥ − 2 Y \geq -2 Y ≥ − 2 B. Domain: − 3 ≤ X ≤ − 2 -3 \leq X \leq -2 − 3 ≤ X ≤ − 2 ; Range: Y ≤ − 2 Y \leq -2 Y ≤ − 2 C. Domain: X ≥ − 3 X \geq -3 X ≥ − 3 ;

When dealing with functions, it's essential to understand the domain and range of the function. The domain of a function is the set of all possible input values (x-values) for which the function is defined, while the range of a function is the set of all possible output values (y-values) that the function can produce.

In this article, we will explore the domain and range of the function $f(x) = -\sqrt{x+3} - 2$. We will analyze the function and determine the domain and range based on the given function.

The Function $f(x) = -\sqrt{x+3} - 2$

The given function is $f(x) = -\sqrt{x+3} - 2$. To determine the domain and range of this function, we need to consider the properties of the square root function.

The square root function is defined only for non-negative real numbers. This means that the expression inside the square root, $x+3$, must be greater than or equal to zero.

Determining the Domain

To determine the domain of the function, we need to find the values of x for which $x+3 \geq 0$. This can be rewritten as $x \geq -3$.

However, we also need to consider the fact that the square root function is not defined for negative real numbers. Therefore, we need to ensure that $x+3$ is not negative.

Combining these two conditions, we can conclude that the domain of the function is $x \geq -3$.

Determining the Range

To determine the range of the function, we need to consider the possible output values of the function.

The function $f(x) = -\sqrt{x+3} - 2$ involves the square root function, which produces non-negative real numbers. However, the negative sign in front of the square root function negates the output value.

Additionally, the constant term $-2$ is subtracted from the output value of the square root function. This means that the output value of the function will always be less than or equal to $-2$.

Therefore, the range of the function is $y \leq -2$.

Conclusion

In conclusion, the domain of the function $f(x) = -\sqrt{x+3} - 2$ is $x \geq -3$, and the range of the function is $y \leq -2$.

This means that the correct answer is:

• Domain: $x \geq -3$
• Range: $y \leq -2$

This answer is consistent with the analysis of the function and the properties of the square root function.

Final Answer

The final answer is:

• Domain: $x \geq -3$
• Range: $y \leq -2$

In the previous article, we explored the domain and range of the function $f(x) = -\sqrt{x+3} - 2$. We determined that the domain of the function is $x \geq -3$ and the range of the function is $y \leq -2$.

In this article, we will answer some frequently asked questions about the domain and range of the function.

Q: What is the domain of the function $f(x) = -\sqrt{x+3} - 2$?

A: The domain of the function $f(x) = -\sqrt{x+3} - 2$ is $x \geq -3$. This means that the function is defined for all real numbers greater than or equal to $-3$.

Q: Why is the domain of the function $x \geq -3$?

A: The domain of the function is $x \geq -3$ because the expression inside the square root, $x+3$, must be greater than or equal to zero. This is a property of the square root function, which is only defined for non-negative real numbers.

Q: What is the range of the function $f(x) = -\sqrt{x+3} - 2$?

A: The range of the function $f(x) = -\sqrt{x+3} - 2$ is $y \leq -2$. This means that the function can produce all real numbers less than or equal to $-2$.

Q: Why is the range of the function $y \leq -2$?

A: The range of the function is $y \leq -2$ because the negative sign in front of the square root function negates the output value, and the constant term $-2$ is subtracted from the output value of the square root function. This means that the output value of the function will always be less than or equal to $-2$.

Q: Can the function $f(x) = -\sqrt{x+3} - 2$ produce any real numbers greater than $-2$?

A: No, the function $f(x) = -\sqrt{x+3} - 2$ cannot produce any real numbers greater than $-2$. The range of the function is $y \leq -2$, which means that the function can only produce real numbers less than or equal to $-2$.

Q: Can the function $f(x) = -\sqrt{x+3} - 2$ produce any real numbers less than $-2$?

A: Yes, the function $f(x) = -\sqrt{x+3} - 2$ can produce any real numbers less than $-2$. The range of the function is $y \leq -2$, which means that the function can produce all real numbers less than or equal to $-2$.

Q: Is the function $f(x) = -\sqrt{x+3} - 2$ a one-to-one function?

A: No, the function $f(x) = -\sqrt{x+3} - 2$ is not a one-to-one function. A one-to-one function is a function that passes the horizontal line test, which means that each output value corresponds to exactly one input value. However, the function $f(x) = -\sqrt{x+3} - 2$ is a decreasing function, which means that it does not pass the horizontal line test.

Q: Is the function $f(x) = -\sqrt{x+3} - 2$ an increasing or decreasing function?

A: The function $f(x) = -\sqrt{x+3} - 2$ is a decreasing function. This means that as the input value increases, the output value decreases.

Conclusion

In conclusion, the domain and range of the function $f(x) = -\sqrt{x+3} - 2$ are $x \geq -3$ and $y \leq -2$, respectively. We also answered some frequently asked questions about the domain and range of the function.