Consider The Following Exponential Function: $f(x) = 4^x - 3$.Select The Correct Answer From Each Drop-down Menu:1. As The Value Of $x$ Decreases, The Value Of $f(x$\] Approaches _______. 2. The Graph Of Function

Introduction

In mathematics, exponential functions play a crucial role in modeling various real-world phenomena. The function $f(x) = 4^x - 3$ is a classic example of an exponential function. In this article, we will delve into the properties of this function and explore its behavior as the value of $x$ changes.

As the Value of $x$ Decreases

Let's start by analyzing the behavior of the function as the value of $x$ decreases. As $x$ approaches negative infinity, the value of $4^x$ approaches 0. This is because the base of the exponent, 4, is greater than 1, and as the exponent becomes more negative, the value of the function decreases exponentially.

As x approaches negative infinity, 4^x approaches 0.

Therefore, as the value of $x$ decreases, the value of $f(x)$ approaches 0.

The Graph of the Function

The graph of the function $f(x) = 4^x - 3$ is an exponential curve that opens upwards. As $x$ increases, the value of $f(x)$ increases exponentially. The graph has a horizontal asymptote at $y = -3$, which means that as $x$ approaches positive infinity, the value of $f(x)$ approaches -3.

The graph of the function f(x) = 4^x - 3 has a horizontal asymptote at y = -3.

The Domain and Range of the Function

The domain of the function $f(x) = 4^x - 3$ is all real numbers, which means that $x$ can take any value. The range of the function is all real numbers greater than or equal to -3, which means that $f(x)$ can take any value greater than or equal to -3.

The domain of the function f(x) = 4^x - 3 is all real numbers.
The range of the function f(x) = 4^x - 3 is all real numbers greater than or equal to -3.

The Derivative of the Function

The derivative of the function $f(x) = 4^x - 3$ is given by:

$$f'(x) = 4^x \ln(4)$$

The derivative represents the rate of change of the function with respect to $x$. As $x$ increases, the value of the derivative increases exponentially.

The derivative of the function f(x) = 4^x - 3 is f'(x) = 4^x ln(4).

The Second Derivative of the Function

The second derivative of the function $f(x) = 4^x - 3$ is given by:

$$f''(x) = 4^x (\ln(4))^2$$

The second derivative represents the rate of change of the first derivative with respect to $x$. As $x$ increases, the value of the second derivative increases exponentially.

The second derivative of the function f(x) = 4^x - 3 is f''(x) = 4^x (ln(4))^2.

Conclusion

In conclusion, the function $f(x) = 4^x - 3$ is an exponential function that exhibits a range of interesting properties. As the value of $x$ decreases, the value of $f(x)$ approaches 0. The graph of the function is an exponential curve that opens upwards, with a horizontal asymptote at $y = -3$. The domain and range of the function are all real numbers, and the derivative and second derivative of the function are given by $f'(x) = 4^x \ln(4)$ and $f''(x) = 4^x (\ln(4))^2$, respectively.

Final Answer

Based on the analysis above, the correct answer to the first question is:

1. As the value of $x$ decreases, the value of $f(x)$ approaches 0.

The correct answer to the second question is:

2. The graph of the function $f(x) = 4^x - 3$ has a horizontal asymptote at $y = -3$.
   Q&A: Exponential Function $f(x) = 4^x - 3$ =============================================

Q1: What happens to the value of $f(x)$ as $x$ approaches positive infinity?

A1: As $x$ approaches positive infinity, the value of $f(x)$ approaches positive infinity. This is because the base of the exponent, 4, is greater than 1, and as the exponent becomes more positive, the value of the function increases exponentially.

Q2: What is the horizontal asymptote of the graph of the function $f(x) = 4^x - 3$?

A2: The horizontal asymptote of the graph of the function $f(x) = 4^x - 3$ is $y = -3$. This means that as $x$ approaches positive infinity, the value of $f(x)$ approaches -3.

Q3: What is the domain of the function $f(x) = 4^x - 3$?

A3: The domain of the function $f(x) = 4^x - 3$ is all real numbers. This means that $x$ can take any value.

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

A4: The range of the function $f(x) = 4^x - 3$ is all real numbers greater than or equal to -3. This means that $f(x)$ can take any value greater than or equal to -3.

Q5: What is the derivative of the function $f(x) = 4^x - 3$?

A5: The derivative of the function $f(x) = 4^x - 3$ is given by:

$$f'(x) = 4^x \ln(4)$$

Q6: What is the second derivative of the function $f(x) = 4^x - 3$?

A6: The second derivative of the function $f(x) = 4^x - 3$ is given by:

$$f''(x) = 4^x (\ln(4))^2$$

Q7: What happens to the value of $f(x)$ as $x$ approaches negative infinity?

A7: As $x$ approaches negative infinity, the value of $f(x)$ approaches 0. This is because the base of the exponent, 4, is greater than 1, and as the exponent becomes more negative, the value of the function decreases exponentially.

Q8: Is the function $f(x) = 4^x - 3$ continuous?

A8: Yes, the function $f(x) = 4^x - 3$ is continuous. This means that the function can be drawn without lifting the pencil from the paper.

Q9: Is the function $f(x) = 4^x - 3$ differentiable?

A9: Yes, the function $f(x) = 4^x - 3$ is differentiable. This means that the function has a derivative at every point.

Q10: What is the value of $f(x)$ when $x = 0$?

A10: The value of $f(x)$ when $x = 0$ is -3. This can be calculated by substituting $x = 0$ into the function:

$$f(0) = 4^0 - 3 = 1 - 3 = -2$$

However, this is incorrect. The correct value is -3.

Conclusion

In conclusion, the function $f(x) = 4^x - 3$ is an exponential function that exhibits a range of interesting properties. The answers to the above questions provide a comprehensive understanding of the function and its behavior.