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 case, we are given the function $f(x) = -3^x - 4$ and asked to find its range. The range of a function is the set of all possible output values it can produce for the given input values. In other words, it is the set of all possible y-values that the function can take.

Understanding the Function

Before we dive into finding the range, let's take a closer look at the function $f(x) = -3^x - 4$. This function involves an exponential term $-3^x$ and a constant term $-4$. The exponential term $-3^x$ is negative because of the negative sign in front of it. As $x$ increases, the value of $-3^x$ will decrease, and as $x$ decreases, the value of $-3^x$ will increase.

Finding the Range

To find the range of the function, we need to consider the possible values of $-3^x$ and then add $-4$ to each of these values. Since $-3^x$ can take on any real value, we can let $-3^x = y$. Then, we have $f(x) = y - 4$. This means that the range of the function $f(x)$ is the set of all possible values of $y - 4$.

Analyzing the Possible Values of $y - 4$

Since $y$ can take on any real value, we can consider the possible values of $y - 4$. If $y$ is positive, then $y - 4$ will be negative. If $y$ is negative, then $y - 4$ will be even more negative. If $y$ is zero, then $y - 4$ will be $-4$. This means that the range of the function $f(x)$ includes all real numbers less than or equal to $-4$.

Conclusion

Based on our analysis, we can conclude that the range of the function $f(x) = -3^x - 4$ is the set of all real numbers less than or equal to $-4$. This means that the correct answer is C. $(-\infty, -4]$. The range of the function is all real numbers less than or equal to $-4$, which is option C.

Final Answer

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

Introduction

In our previous article, we discussed the range of the function $f(x) = -3^x - 4$. We concluded that the range of the function is the set of all real numbers less than or equal to $-4$. In this article, we will provide a Q&A section to further clarify any doubts and provide additional information.

Q&A

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

A1: The range of the function $f(x) = -3^x - 4$ is the set of all real numbers less than or equal to $-4$. This means that the function can take on any value less than or equal to $-4$.

Q2: Why is the range of the function $f(x) = -3^x - 4$ all real numbers less than or equal to $-4$?

A2: The range of the function $f(x) = -3^x - 4$ is all real numbers less than or equal to $-4$ because the exponential term $-3^x$ can take on any real value. When we add $-4$ to this value, we get a value that is less than or equal to $-4$.

Q3: Can the function $f(x) = -3^x - 4$ take on any value greater than $-4$?

A3: No, the function $f(x) = -3^x - 4$ cannot take on any value greater than $-4$. This is because the exponential term $-3^x$ can take on any real value, but when we add $-4$ to this value, we get a value that is less than or equal to $-4$.

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

A4: The domain of the function $f(x) = -3^x - 4$ is all real numbers. This means that the function can take on any real value for the input $x$.

Q5: Can the function $f(x) = -3^x - 4$ be represented in a different form?

A5: Yes, the function $f(x) = -3^x - 4$ can be represented in a different form. We can let $y = -3^x$ and then solve for $x$ in terms of $y$. This will give us a different representation of the function.

Conclusion

In this Q&A article, we have provided additional information and clarification on the range of the function $f(x) = -3^x - 4$. We have answered questions such as what the range of the function is, why the range is all real numbers less than or equal to $-4$, and whether the function can take on any value greater than $-4$. We have also discussed the domain of the function and whether it can be represented in a different form.

Final Answer

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