What Is The Range Of The Function F ( X ) = − 3 X − 4 F(x) = -3^x - 4 F ( X ) = − 3 X − 4 ?A. ( − 4 , ∞ (-4, \infty ( − 4 , ∞ ] B. ( − ∞ , − 3 (-\infty, -3 ( − ∞ , − 3 ] C. ( − ∞ , − 4 (-\infty, -4 ( − ∞ , − 4 ] D. ( − 3 , ∞ (-3, \infty ( − 3 , ∞ ]

Introduction

When dealing with functions, understanding the range is crucial in determining the possible output values. In this article, we will explore the range of the function $f(x) = -3^x - 4$. The range of a function is the set of all possible output values it can produce for the given input values. To find the range, we need to analyze the behavior of the function as $x$ varies.

Understanding the Function

The given function is $f(x) = -3^x - 4$. This is an exponential function with a negative base, which means it will decrease as $x$ increases. The negative sign in front of the exponent indicates that the function will be negative for all values of $x$. The constant term $-4$ shifts the function downwards, making it even more negative.

Analyzing the Behavior of the Function

To find the range of the function, we need to analyze its behavior as $x$ varies. Let's start by finding the domain of the function. Since the function involves an exponential term, it is defined for all real values of $x$. Now, let's consider the behavior of the function as $x$ approaches positive infinity and negative infinity.

Behavior as $x$ Approaches Positive Infinity

As $x$ approaches positive infinity, the term $-3^x$ approaches negative infinity. This is because the exponential function $3^x$ grows rapidly as $x$ increases, and the negative sign in front of it makes it negative. Therefore, as $x$ approaches positive infinity, the function $f(x) = -3^x - 4$ approaches negative infinity.

Behavior as $x$ Approaches Negative Infinity

As $x$ approaches negative infinity, the term $-3^x$ approaches positive infinity. This is because the exponential function $3^x$ grows rapidly as $x$ decreases, and the negative sign in front of it makes it positive. Therefore, as $x$ approaches negative infinity, the function $f(x) = -3^x - 4$ approaches positive infinity.

Finding the Range

Now that we have analyzed the behavior of the function as $x$ varies, we can find the range. Since the function approaches negative infinity as $x$ approaches positive infinity and approaches positive infinity as $x$ approaches negative infinity, the range of the function is all real numbers less than or equal to $-4$. In other words, the range of the function $f(x) = -3^x - 4$ is $(-\infty, -4]$.

Conclusion

In conclusion, the range of the function $f(x) = -3^x - 4$ is $(-\infty, -4]$. This means that the function can produce all real numbers less than or equal to $-4$ as output values. The function is defined for all real values of $x$ and approaches negative infinity as $x$ approaches positive infinity and approaches positive infinity as $x$ approaches negative infinity.

Final Answer

The final answer is $(-\infty, -4]$.

Introduction

In our previous article, we explored the range of the function $f(x) = -3^x - 4$. We analyzed the behavior of the function as $x$ varies and found that the range of the function is $(-\infty, -4]$. In this article, we will answer some frequently asked questions related to the range of the function.

Q&A

Q1: What is the domain of the function $f(x) = -3^x - 4$?

A1: The domain of the function $f(x) = -3^x - 4$ is all real numbers, since the function involves an exponential term and is defined for all real values of $x$.

Q2: How does the function behave as $x$ approaches positive infinity?

A2: As $x$ approaches positive infinity, the term $-3^x$ approaches negative infinity. This is because the exponential function $3^x$ grows rapidly as $x$ increases, and the negative sign in front of it makes it negative.

Q3: How does the function behave as $x$ approaches negative infinity?

A3: As $x$ approaches negative infinity, the term $-3^x$ approaches positive infinity. This is because the exponential function $3^x$ grows rapidly as $x$ decreases, and the negative sign in front of it makes it positive.

Q4: What is the range of the function $f(x) = -3^x - 4$?

A4: The range of the function $f(x) = -3^x - 4$ is $(-\infty, -4]$. This means that the function can produce all real numbers less than or equal to $-4$ as output values.

Q5: Why is the range of the function $f(x) = -3^x - 4$ limited to $(-\infty, -4]$?

A5: The range of the function $f(x) = -3^x - 4$ is limited to $(-\infty, -4]$ because the function approaches negative infinity as $x$ approaches positive infinity and approaches positive infinity as $x$ approaches negative infinity. This means that the function can never produce values greater than $-4$.

Q6: Can the function $f(x) = -3^x - 4$ produce values greater than $-4$?

A6: No, the function $f(x) = -3^x - 4$ cannot produce values greater than $-4$. This is because the function approaches negative infinity as $x$ approaches positive infinity and approaches positive infinity as $x$ approaches negative infinity.

Q7: What is the significance of the constant term $-4$ in the function $f(x) = -3^x - 4$?

A7: The constant term $-4$ in the function $f(x) = -3^x - 4$ shifts the function downwards, making it even more negative. This means that the function will always produce values less than or equal to $-4$.

Conclusion

In conclusion, the range of the function $f(x) = -3^x - 4$ is $(-\infty, -4]$. This means that the function can produce all real numbers less than or equal to $-4$ as output values. We hope that this Q&A article has helped to clarify any doubts you may have had about the range of the function.

Final Answer

The final answer is $(-\infty, -4]$.