Consider The Graph Of The Function F ( X ) = − ( 2 ) X + 4 F(x) = -(2)^x + 4 F ( X ) = − ( 2 ) X + 4 .What Is The Range Of Function F F F ?A. { Y ∣ − ∞ \textless Y \textless 0 } \{y \mid -\infty \ \textless \ Y \ \textless \ 0\} { Y ∣ − ∞ \textless Y \textless 0 } B. { Y ∣ − ∞ \textless Y \textless 4 } \{y \mid -\infty \ \textless \ Y \ \textless \ 4\} { Y ∣ − ∞ \textless Y \textless 4 } C.

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Introduction

In mathematics, a function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. The graph of a function is a visual representation of the function, showing the relationship between the input and output values. In this article, we will consider the graph of the function f(x)=−(2)x+4f(x) = -(2)^x + 4 and determine its range.

The Function f(x)=−(2)x+4f(x) = -(2)^x + 4

The function f(x)=−(2)x+4f(x) = -(2)^x + 4 is an exponential function with a base of 2 and a coefficient of -1. The function is defined for all real numbers xx. To understand the graph of this function, we need to analyze its behavior as xx varies.

Asymptotes

The function f(x)=−(2)x+4f(x) = -(2)^x + 4 has a horizontal asymptote at y=4y = 4. This means that as xx approaches infinity, the value of f(x)f(x) approaches 4. Similarly, as xx approaches negative infinity, the value of f(x)f(x) approaches negative infinity.

Domain and Range

The domain of the function f(x)=−(2)x+4f(x) = -(2)^x + 4 is all real numbers xx. The range of the function is the set of all possible output values, which we need to determine.

Determining the Range

To determine the range of the function, we need to analyze its behavior as xx varies. We can start by finding the minimum value of the function.

Finding the Minimum Value

The minimum value of the function f(x)=−(2)x+4f(x) = -(2)^x + 4 occurs when the exponential term −(2)x-(2)^x is at its maximum value. This happens when x=−∞x = -\infty, since the exponential function −(2)x-(2)^x approaches 0 as xx approaches negative infinity.

Calculating the Minimum Value

To calculate the minimum value of the function, we substitute x=−∞x = -\infty into the function:

f(−∞)=−(2)−∞+4f(-\infty) = -(2)^{-\infty} + 4

Since −(2)−∞-(2)^{-\infty} approaches 0 as xx approaches negative infinity, we have:

f(−∞)=0+4=4f(-\infty) = 0 + 4 = 4

Conclusion

The minimum value of the function f(x)=−(2)x+4f(x) = -(2)^x + 4 is 4, which occurs when x=−∞x = -\infty. Since the function approaches negative infinity as xx approaches positive infinity, the range of the function is the set of all values between 4 and negative infinity.

The Range of the Function

The range of the function f(x)=−(2)x+4f(x) = -(2)^x + 4 is:

{y∣−∞ \textless y \textless 4}\{y \mid -\infty \ \textless \ y \ \textless \ 4\}

This is option B in the discussion category.

Final Answer

Introduction

In our previous article, we discussed the graph of the function f(x)=−(2)x+4f(x) = -(2)^x + 4 and determined its range. In this article, we will answer some frequently asked questions about the graph of this function.

Q: What is the domain of the function f(x)=−(2)x+4f(x) = -(2)^x + 4?

A: The domain of the function f(x)=−(2)x+4f(x) = -(2)^x + 4 is all real numbers xx. This means that the function is defined for all values of xx.

Q: What is the range of the function f(x)=−(2)x+4f(x) = -(2)^x + 4?

A: The range of the function f(x)=−(2)x+4f(x) = -(2)^x + 4 is the set of all values between 4 and negative infinity. This is because the function approaches negative infinity as xx approaches positive infinity, and the minimum value of the function is 4.

Q: What is the horizontal asymptote of the function f(x)=−(2)x+4f(x) = -(2)^x + 4?

A: The horizontal asymptote of the function f(x)=−(2)x+4f(x) = -(2)^x + 4 is y=4y = 4. This means that as xx approaches infinity, the value of f(x)f(x) approaches 4.

Q: What is the minimum value of the function f(x)=−(2)x+4f(x) = -(2)^x + 4?

A: The minimum value of the function f(x)=−(2)x+4f(x) = -(2)^x + 4 is 4. This occurs when x=−∞x = -\infty, since the exponential term −(2)x-(2)^x approaches 0 as xx approaches negative infinity.

Q: How does the function f(x)=−(2)x+4f(x) = -(2)^x + 4 behave as xx approaches positive infinity?

A: As xx approaches positive infinity, the value of f(x)f(x) approaches negative infinity. This is because the exponential term −(2)x-(2)^x grows without bound as xx increases.

Q: Is the function f(x)=−(2)x+4f(x) = -(2)^x + 4 continuous?

A: Yes, the function f(x)=−(2)x+4f(x) = -(2)^x + 4 is continuous. This means that the function has no gaps or jumps in its graph.

Q: Is the function f(x)=−(2)x+4f(x) = -(2)^x + 4 differentiable?

A: Yes, the function f(x)=−(2)x+4f(x) = -(2)^x + 4 is differentiable. This means that the function has a derivative at every point in its domain.

Conclusion

In this article, we answered some frequently asked questions about the graph of the function f(x)=−(2)x+4f(x) = -(2)^x + 4. We discussed the domain and range of the function, as well as its horizontal asymptote and minimum value. We also explored how the function behaves as xx approaches positive infinity and whether the function is continuous and differentiable.

Final Answer

The final answer is that the range of the function f(x)=−(2)x+4f(x) = -(2)^x + 4 is {y∣−∞ \textless y \textless 4}\{y \mid -\infty \ \textless \ y \ \textless \ 4\}.