Problem 16 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether the equation represents exponential growth, exponential decay, or neither. Explain. \(y=16.5(1.025)^{\frac{1}{x}}\)

Step-by-Step Solution

Answer

Neither; the exponent leads to a constant value as \(x\) becomes large.

Step 1: Identify the Form of an Exponential Equation

An exponential equation is generally in the form \(y = ab^x\), where \(a\) is a constant, \(b\) is the base of the exponential function, and \(x\) is the exponent. Our given equation is \(y=16.5(1.025)^{\frac{1}{x}}\).

Step 2: Analyze the Exponent

In our equation, the exponent is \(\frac{1}{x}\), which is not a simple \(x\) variable. This creates a different behavior than typical simple exponential functions where the exponent is just \(x\).

Step 3: Determine Effect of the Exponent on the Base

As \(x\) becomes very large, \(\frac{1}{x}\) becomes very small, approaching zero. This would make \((1.025)^{\frac{1}{x}}\) approach \(1\), leading \(y\) towards a constant value of \(16.5\).

Step 4: Classify the Equation

Because the exponent \(\frac{1}{x}\) leads to neither consistent growth nor decay over all positive \(x\), the behavior of the function does not fit exponential growth or decay definitions. The function trends towards a constant value instead of consistently increasing or decreasing.

Key Concepts

Exponential Growth

Exponential growth is a fascinating concept often encountered in real-world phenomena such as population increase, investment growth, and bacterial multiplication. In mathematical terms, exponential growth occurs when the increase in a quantity is proportional to its current value. This is usually represented by the equation \(y = a \cdot b^x\), where \(a\) is a constant, and \(b > 1\) indicates growth. As \(x\) increases, the value of \(y\) grows rapidly, depicting an accelerating rise.

• The base \(b\) is greater than 1, which ensures that the function increases.
• The exponent \(x\) plays a crucial role, as it is responsible for the rapid growth.
• This kind of growth is famously known as "compounded growth."

Understanding exponential growth involves acknowledging that each step forward multiplies the previous quantity by a constant factor. This characteristic is what differentiates it from linear growth, where increases are constant in absolute terms.

Exponential Decay

Exponential decay is essentially the inverse of exponential growth. Here, the decrease is proportional to the current value of the quantity. The mathematical model is similar: \(y = a \cdot b^x\), but with a crucial difference — the base \(b\) is less than 1 (typically, \(0 < b < 1\)). This choice of base ensures that as \(x\) increases, \(y\) decreases gradually, demonstrating a form of decelerating shrinkage over time.

• The base of the function \(b\) ensures that each increase in \(x\) results in a reduction of \(y\).
• It is commonly observed in natural phenomena like radioactive decay and cooling processes.
• This type of function portrays a "halving" behavior that is often predictable.

Exponential decay describes how a variable decreases rapidly at first and then levels off. Understanding this concept can be key in fields like half-life calculations in Chemistry, where a reactive substance diminishes by a certain percentage over consistent timespan intervals.

Mathematical Analysis

Mathematical analysis explores functions deeply to understand their behaviors and properties through precise mathematical definitions and techniques. It involves several fundamental tools such as limits, derivatives, and integrals. Analyzing functions allows us to predict how they behave across different conditions.
For the given equation \(y=16.5(1.025)^{\frac{1}{x}}\), analysis was crucial:

• First, understanding the form: It didn't fit the standard \(y = ab^x\) format because of its exponent \(\frac{1}{x}\).
• Next, by applying limits, analyzing how \(\frac{1}{x}\) behaves as \(x\) increases or decreases gave insights.
• Here, \(\frac{1}{x}\) tends to zero as \(x\) grows, causing the base raised to this power to approach 1.

Mathematical analysis offers more than just solving equations — it provides a framework to interpret how various parts of an equation interact, leading to broader insights and deeper understanding of mathematical phenomena.