Determine The Range Of $y = A B^x$ With An Initial Value Of 5 And A Decay Of $7\%$.A) All Real Numbers B) \$y \ \textgreater \ 0$[/tex\] C) $y \ \textless \ 0$ D) $y \leq 0$

Introduction

Exponential functions are a fundamental concept in mathematics, used to model various real-world phenomena, such as population growth, chemical reactions, and financial investments. In this article, we will explore the range of an exponential function with a decay of 7%, given an initial value of 5. We will analyze the function, determine its behavior, and identify the possible range of values it can take.

Understanding Exponential Functions

An exponential function is a mathematical function of the form $y = ab^x$, where $a$ is the initial value, $b$ is the base, and $x$ is the exponent. The base $b$ determines the rate of growth or decay of the function. If $b > 1$, the function grows exponentially, while if $0 < b < 1$, the function decays exponentially.

The Given Function

In this case, we are given the function $y = ab^x$ with an initial value of $a = 5$ and a decay of $7\%$. This means that the base $b$ is less than 1, since a decay of 7% implies a reduction in the value of the function over time.

Analyzing the Function

To determine the range of the function, we need to analyze its behavior as $x$ varies. Since the function decays exponentially, we can expect the value of $y$ to decrease as $x$ increases.

The Decay Rate

The decay rate is given by the base $b$, which is less than 1. Specifically, the decay rate is $b = 1 - 0.07 = 0.93$. This means that for every unit increase in $x$, the value of $y$ decreases by 7%.

The Range of the Function

To determine the range of the function, we need to consider the possible values of $y$ as $x$ varies. Since the function decays exponentially, we can expect the value of $y$ to approach 0 as $x$ increases.

Option A: All Real Numbers

Option A suggests that the range of the function is all real numbers. However, this is not possible, since the function decays exponentially and approaches 0 as $x$ increases.

Option B: $y > 0$

Option B suggests that the range of the function is all positive real numbers. However, this is not possible, since the function decays exponentially and approaches 0 as $x$ increases.

Option C: $y < 0$

Option C suggests that the range of the function is all negative real numbers. However, this is not possible, since the function decays exponentially and approaches 0 as $x$ increases.

Option D: $y \leq 0$

Option D suggests that the range of the function is all non-positive real numbers, including 0. This is the correct answer, since the function decays exponentially and approaches 0 as $x$ increases.

Conclusion

In conclusion, the range of the exponential function $y = ab^x$ with an initial value of 5 and a decay of 7% is all non-positive real numbers, including 0. This is because the function decays exponentially and approaches 0 as $x$ increases.

References

• [1] "Exponential Functions" by Math Is Fun
• [2] "Exponential Decay" by Khan Academy

Final Answer

Introduction

In our previous article, we explored the range of an exponential function with a decay of 7%, given an initial value of 5. We analyzed the function, determined its behavior, and identified the possible range of values it can take. In this article, we will answer some frequently asked questions related to the topic.

Q: What is the difference between an exponential growth function and an exponential decay function?

A: An exponential growth function is a function of the form $y = ab^x$, where $a$ is the initial value and $b$ is the base, with $b > 1$. This type of function grows exponentially, meaning that the value of $y$ increases rapidly as $x$ increases. On the other hand, an exponential decay function is a function of the form $y = ab^x$, where $a$ is the initial value and $b$ is the base, with $0 < b < 1$. This type of function decays exponentially, meaning that the value of $y$ decreases rapidly as $x$ increases.

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

A: To determine the range of an exponential function, you need to analyze its behavior as $x$ varies. If the function grows exponentially, the range will be all real numbers greater than or equal to the initial value. If the function decays exponentially, the range will be all real numbers less than or equal to the initial value.

Q: What is the significance of the base $b$ in an exponential function?

A: The base $b$ determines the rate of growth or decay of the function. If $b > 1$, the function grows exponentially. If $0 < b < 1$, the function decays exponentially. The value of $b$ also determines the steepness of the function's graph.

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

A: No, an exponential function cannot have a range of all real numbers. If the function grows exponentially, the range will be all real numbers greater than or equal to the initial value. If the function decays exponentially, the range will be all real numbers less than or equal to the initial value.

Q: How do I determine the range of an exponential function with a decay of 7%?

A: To determine the range of an exponential function with a decay of 7%, you need to analyze its behavior as $x$ varies. Since the function decays exponentially, the range will be all real numbers less than or equal to the initial value. In this case, the initial value is 5, so the range will be all real numbers less than or equal to 5.

Q: Can an exponential function have a range of all non-positive real numbers?

A: Yes, an exponential function can have a range of all non-positive real numbers. This is the case when the function decays exponentially and approaches 0 as $x$ increases.

Conclusion

In conclusion, determining the range of an exponential function with decay requires analyzing its behavior as $x$ varies. The base $b$ determines the rate of growth or decay of the function, and the initial value determines the range of the function. We hope this Q&A article has provided you with a better understanding of the topic.

References

• [1] "Exponential Functions" by Math Is Fun
• [2] "Exponential Decay" by Khan Academy

Final Answer

The final answer is: $\boxed{D}$