The Range Of $f(x)=7 \cdot 4^x$ Is All Positive Real Numbers.A. True B. False

Introduction

Exponential functions are a fundamental concept in mathematics, and understanding their behavior is crucial for solving various mathematical problems. In this article, we will delve into the range of exponential functions, specifically the function $f(x) = 7 \cdot 4^x$. We will analyze the properties of this function and determine whether its range is all positive real numbers.

What is an Exponential Function?

An exponential function is a mathematical function of the form $f(x) = a \cdot b^x$, where $a$ and $b$ are constants, and $b$ is a positive real number. The base $b$ determines the growth rate of the function, and the exponent $x$ determines the input value. Exponential functions are characterized by their rapid growth or decay, depending on the base and the exponent.

Properties of Exponential Functions

Exponential functions have several important properties that are essential for understanding their behavior. Some of these properties include:

• Domain: The domain of an exponential function is all real numbers, unless the base is negative, in which case the domain is restricted to non-negative real numbers.
• Range: The range of an exponential function depends on the base and the exponent. If the base is positive, the range is all positive real numbers. If the base is negative, the range is all negative real numbers.
• One-to-One: Exponential functions are one-to-one functions, meaning that each output value corresponds to a unique input value.
• Continuous: Exponential functions are continuous functions, meaning that they can be graphed without lifting the pencil from the paper.

The Function $f(x) = 7 \cdot 4^x$

The function $f(x) = 7 \cdot 4^x$ is an exponential function with a base of 4 and a coefficient of 7. Since the base is positive, the function is increasing for all real values of $x$. The coefficient 7 scales the function, making it larger than the function $f(x) = 4^x$.

Analyzing the Range of $f(x) = 7 \cdot 4^x$

To determine the range of $f(x) = 7 \cdot 4^x$, we need to consider the possible values of the function for different input values of $x$. Since the function is increasing for all real values of $x$, we can conclude that the range is all positive real numbers.

Proof of the Range

To prove that the range of $f(x) = 7 \cdot 4^x$ is all positive real numbers, we can use the following argument:

• For any positive real number $y$, we can find a real number $x$ such that $f(x) = y$. Specifically, we can choose $x = \log_4 \left( \frac{y}{7} \right)$, which satisfies the equation $f(x) = 7 \cdot 4^x = y$.
• Since we can find a real number $x$ for any positive real number $y$, the range of $f(x) = 7 \cdot 4^x$ is all positive real numbers.

Conclusion

In conclusion, the range of the function $f(x) = 7 \cdot 4^x$ is all positive real numbers. This is because the function is increasing for all real values of $x$, and we can find a real number $x$ for any positive real number $y$. Therefore, the correct answer is:

A. True

Final Thoughts

Exponential functions are a fundamental concept in mathematics, and understanding their behavior is crucial for solving various mathematical problems. In this article, we analyzed the range of the function $f(x) = 7 \cdot 4^x$ and determined that it is all positive real numbers. We also provided a proof of the range using the properties of exponential functions. We hope that this article has provided a comprehensive understanding of the range of exponential functions and has helped readers to develop a deeper appreciation for the beauty and power of mathematics.

Introduction

In our previous article, we explored the range of exponential functions, specifically the function $f(x) = 7 \cdot 4^x$. We analyzed the properties of this function and determined that its range is all positive real numbers. In this article, we will address some of the most frequently asked questions related to the range of exponential functions.

Q&A

Q: What is the range of the function $f(x) = 2^x$?

A: The range of the function $f(x) = 2^x$ is all positive real numbers. This is because the base 2 is positive, and the function is increasing for all real values of $x$.

Q: How do I determine the range of an exponential function?

A: To determine the range of an exponential function, you need to consider the base and the exponent. If the base is positive, the range is all positive real numbers. If the base is negative, the range is all negative real numbers.

Q: What is the difference between the range and the domain of an exponential function?

A: The domain of an exponential function is the set of all input values for which the function is defined. The range of an exponential function is the set of all output values that the function can produce. In other words, the domain is the input values, and the range is the output values.

Q: Can an exponential function have a range that is not all positive real numbers?

A: Yes, an exponential function can have a range that is not all positive real numbers. For example, the function $f(x) = -2^x$ has a range that is all negative real numbers.

Q: How do I find the range of a composite function?

A: To find the range of a composite function, you need to consider the range of each individual function in the composition. For example, if you have a composite function $f(g(x))$, you need to find the range of $g(x)$ and then plug that range into $f(x)$ to find the final range.

Q: Can an exponential function have a range that is all real numbers?

A: Yes, an exponential function can have a range that is all real numbers. For example, the function $f(x) = e^x$ has a range that is all real numbers.

Q: How do I determine if an exponential function is increasing or decreasing?

A: To determine if an exponential function is increasing or decreasing, you need to consider the base and the exponent. If the base is positive, the function is increasing for all real values of $x$. If the base is negative, the function is decreasing for all real values of $x$.

Q: Can an exponential function have a range that is not continuous?

A: No, an exponential function cannot have a range that is not continuous. Exponential functions are continuous functions, meaning that they can be graphed without lifting the pencil from the paper.

Conclusion

In conclusion, the range of exponential functions is a fundamental concept in mathematics, and understanding it is crucial for solving various mathematical problems. We hope that this article has provided a comprehensive understanding of the range of exponential functions and has helped readers to develop a deeper appreciation for the beauty and power of mathematics.

Final Thoughts

Exponential functions are a fundamental concept in mathematics, and understanding their behavior is crucial for solving various mathematical problems. In this article, we addressed some of the most frequently asked questions related to the range of exponential functions. We hope that this article has provided a comprehensive understanding of the range of exponential functions and has helped readers to develop a deeper appreciation for the beauty and power of mathematics.