Analyzing Exponential Function F(x) = (1.07)^x Growth And Decay

In the realm of mathematics, exponential functions play a pivotal role in modeling phenomena that exhibit rapid growth or decay. These functions are characterized by a constant base raised to a variable exponent, and their behavior is dictated by the value of the base. Let's delve into the intricacies of exponential functions and explore how to interpret their significance.

Dissecting the Function f(x) = (1.07)^x

The function presented, f(x) = (1.07)^x, is a classic example of an exponential function. To decipher its meaning, we need to carefully examine its components. The base of this function is 1.07, and the exponent is the variable x. The base is the key to understanding whether the function represents growth or decay.

To determine if the function represents exponential growth or decay, we need to analyze the value of the base. In this case, the base is 1.07. A base greater than 1 signifies exponential growth, while a base between 0 and 1 indicates exponential decay. Since 1.07 is greater than 1, the function f(x) = (1.07)^x represents exponential growth. The base of the exponential function, 1.07, holds the key to understanding the rate of growth. To determine the growth rate, we subtract 1 from the base and express the result as a percentage. In this case, 1.07 - 1 = 0.07, which translates to a 7% growth rate. Therefore, the function f(x) = (1.07)^x represents exponential growth at a rate of 7% for each unit increase in x. The concept of exponential growth is fundamental in various fields, including finance, biology, and physics. It describes situations where a quantity increases at a rate proportional to its current value. For instance, compound interest, population growth, and the spread of infectious diseases often exhibit exponential growth patterns. Understanding exponential growth allows us to make predictions and model real-world phenomena effectively.

Exponential Growth Unveiled

Exponential growth occurs when a quantity increases at a rate proportional to its current value. This means that the larger the quantity, the faster it grows. Exponential growth is often observed in situations where there are abundant resources and minimal constraints. A classic example of exponential growth is the growth of a bacterial colony in a petri dish. Initially, the bacteria reproduce slowly, but as their numbers increase, the rate of reproduction accelerates, leading to a rapid increase in population. In mathematical terms, exponential growth is represented by a function of the form f(x) = a(1 + r)^x, where a is the initial amount, r is the growth rate, and x is the time period. The base of the exponential term, (1 + r), determines the rate of growth. A base greater than 1 indicates exponential growth, while a base between 0 and 1 indicates exponential decay. Exponential growth has profound implications in various fields, including finance, economics, and environmental science. Understanding exponential growth helps us make informed decisions about investments, resource management, and public policy.

Exponential Decay Explained

Conversely, exponential decay occurs when a quantity decreases at a rate proportional to its current value. This means that the smaller the quantity, the slower it decays. Exponential decay is commonly observed in situations where a substance is breaking down or a population is declining. A typical example of exponential decay is the radioactive decay of an unstable isotope. The rate of decay is constant, meaning that the fraction of the isotope that decays in a given time period remains the same regardless of the amount present. In mathematical terms, exponential decay is represented by a function of the form f(x) = a(1 - r)^x, where a is the initial amount, r is the decay rate, and x is the time period. The base of the exponential term, (1 - r), determines the rate of decay. A base between 0 and 1 indicates exponential decay. Exponential decay is a fundamental concept in various fields, including nuclear physics, chemistry, and pharmacology. Understanding exponential decay helps us predict the behavior of radioactive materials, the rate of chemical reactions, and the elimination of drugs from the body.

Deciphering the Growth Rate

In the function f(x) = (1.07)^x, the base 1.07 signifies a growth factor. To determine the growth rate as a percentage, we subtract 1 from the base and multiply by 100. This yields (1.07 - 1) * 100 = 7%. Therefore, the function represents an exponential growth of 7%. The growth rate is a crucial parameter in exponential functions as it quantifies the rate at which the quantity is increasing or decreasing. A higher growth rate indicates a more rapid increase, while a lower growth rate indicates a slower increase or even a decrease. The growth rate is expressed as a percentage, making it easier to compare the growth rates of different exponential functions. In financial applications, the growth rate is often referred to as the interest rate or the rate of return. In population studies, the growth rate indicates the percentage increase in population size over a specific period. Understanding the growth rate is essential for making predictions and informed decisions in various fields.

Why 93% is Incorrect

The options suggesting a 93% change are incorrect because they misinterpret the exponential nature of the function. The function does not represent a one-time change of 93%. Instead, it represents a continuous growth of 7% for each unit increase in x. The key to understanding exponential functions lies in recognizing that the growth or decay is not linear but rather proportional to the current value. This means that the change in the quantity is not constant but rather increases or decreases over time. In the case of exponential growth, the quantity increases at an accelerating rate, while in the case of exponential decay, the quantity decreases at a decelerating rate. The options suggesting a 93% change fail to capture this dynamic nature of exponential functions. It's crucial to focus on the base of the function, which directly determines the growth or decay rate, rather than interpreting it as a simple percentage change.

Conclusion: The Verdict

Therefore, the function f(x) = (1.07)^x unequivocally represents exponential growth of 7%. This understanding is crucial for interpreting various real-world phenomena that exhibit exponential behavior. Exponential functions are a powerful tool for modeling growth and decay in various fields, including finance, biology, and physics. Mastering the concepts of exponential growth and decay allows us to make predictions, analyze trends, and gain insights into the dynamics of complex systems. By understanding the role of the base and the growth rate, we can effectively interpret exponential functions and apply them to solve real-world problems. The function f(x) = (1.07)^x serves as a fundamental example of exponential growth, highlighting the importance of understanding these concepts in mathematics and beyond.

Therefore, the correct answer is A. Exponential growth of 7%.