What Is The Range Of The Function F ( X ) = − 2 ( 6 X ) + 3 F(x) = -2\left(6^x\right) + 3 F ( X ) = − 2 ( 6 X ) + 3 ?A. { (-∞, -2]$}$ B. { (-∞, 3)$}$ C. { [-2, ∞)$}$ D. { [3, ∞)$}$

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Introduction

When dealing with functions, understanding the range is crucial in mathematics. 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 will explore the range of the function f(x)=−2(6x)+3f(x) = -2\left(6^x\right) + 3. We will analyze the function, identify its key characteristics, and determine its range.

Understanding the Function

The given function is f(x)=−2(6x)+3f(x) = -2\left(6^x\right) + 3. This function involves an exponential term, 6x6^x, which is multiplied by −2-2 and then added to 33. To understand the behavior of this function, let's break it down into its components.

Exponential Term

The exponential term, 6x6^x, is a key component of the function. This term represents an exponential growth or decay, depending on the value of xx. When xx is positive, the term 6x6^x grows exponentially, and when xx is negative, the term 6x6^x decays exponentially.

Multiplication by −2-2

The term −2-2 is multiplied by the exponential term, which affects the growth or decay of the function. Multiplying by a negative number inverts the direction of the growth or decay, resulting in a downward or upward trend.

Addition of 33

The final component of the function is the addition of 33. This term shifts the function upward or downward, depending on the value of xx.

Analyzing the Function

To determine the range of the function, we need to analyze its behavior as xx varies. Let's consider the following cases:

Case 1: x→−∞x \to -\infty

As xx approaches negative infinity, the exponential term 6x6^x approaches 00. Therefore, the function f(x)=−2(6x)+3f(x) = -2\left(6^x\right) + 3 approaches 33 as xx approaches negative infinity.

Case 2: x→∞x \to \infty

As xx approaches positive infinity, the exponential term 6x6^x grows exponentially. Therefore, the function f(x)=−2(6x)+3f(x) = -2\left(6^x\right) + 3 approaches negative infinity as xx approaches positive infinity.

Case 3: x=0x = 0

When x=0x = 0, the exponential term 6x6^x is equal to 11. Therefore, the function f(x)=−2(6x)+3f(x) = -2\left(6^x\right) + 3 is equal to −2(1)+3=1-2(1) + 3 = 1 when x=0x = 0.

Determining the Range

Based on the analysis of the function, we can determine its range. The function approaches 33 as xx approaches negative infinity, and it approaches negative infinity as xx approaches positive infinity. Therefore, the range of the function is the set of all values between 33 and negative infinity.

Conclusion

In conclusion, the range of the function f(x)=−2(6x)+3f(x) = -2\left(6^x\right) + 3 is the set of all values between 33 and negative infinity. This range is represented by the interval (−∞,3)(-\infty, 3).

Final Answer

The final answer is: B\boxed{B}

Introduction

In our previous article, we explored the range of the function f(x)=−2(6x)+3f(x) = -2\left(6^x\right) + 3. We analyzed the function, identified its key characteristics, and determined its range. In this article, we will answer some frequently asked questions related to the range of the function.

Q&A

Q: What is the range of the function f(x)=−2(6x)+3f(x) = -2\left(6^x\right) + 3?

A: The range of the function f(x)=−2(6x)+3f(x) = -2\left(6^x\right) + 3 is the set of all values between 33 and negative infinity. This range is represented by the interval (−∞,3)(-\infty, 3).

Q: Why does the function approach 33 as xx approaches negative infinity?

A: The function approaches 33 as xx approaches negative infinity because the exponential term 6x6^x approaches 00 as xx approaches negative infinity. Therefore, the function f(x)=−2(6x)+3f(x) = -2\left(6^x\right) + 3 approaches 33 as xx approaches negative infinity.

Q: Why does the function approach negative infinity as xx approaches positive infinity?

A: The function approaches negative infinity as xx approaches positive infinity because the exponential term 6x6^x grows exponentially as xx approaches positive infinity. Therefore, the function f(x)=−2(6x)+3f(x) = -2\left(6^x\right) + 3 approaches negative infinity as xx approaches positive infinity.

Q: What is the value of the function when x=0x = 0?

A: When x=0x = 0, the exponential term 6x6^x is equal to 11. Therefore, the function f(x)=−2(6x)+3f(x) = -2\left(6^x\right) + 3 is equal to −2(1)+3=1-2(1) + 3 = 1 when x=0x = 0.

Q: How can I determine the range of a function like f(x)=−2(6x)+3f(x) = -2\left(6^x\right) + 3?

A: To determine the range of a function like f(x)=−2(6x)+3f(x) = -2\left(6^x\right) + 3, you need to analyze the function's behavior as xx varies. You should consider the following cases:

  • x→−∞x \to -\infty
  • x→∞x \to \infty
  • x=0x = 0

By analyzing these cases, you can determine the range of the function.

Conclusion

In conclusion, the range of the function f(x)=−2(6x)+3f(x) = -2\left(6^x\right) + 3 is the set of all values between 33 and negative infinity. This range is represented by the interval (−∞,3)(-\infty, 3). We hope that this Q&A article has helped you understand the range of the function and how to determine the range of similar functions.

Final Answer

The final answer is: B\boxed{B}