Problem 20 - Algebra 2

Question

Without graphing, determine whether each function represents exponential growth or exponential decay. $$ f(x)=4\left(\frac{5}{6}\right)^{x} $$

Step-by-Step Solution

Answer

The function \(f(x)=4\left(\frac{5}{6}\right)^{x}\) represents exponential decay.

Step 1: Identify the base of the exponential function

The given function is \(f(x)=4\left(\frac{5}{6}\right)^{x}\). Here, the base of the power is \(\frac{5}{6}\).

Step 2: Determine whether the base indicates growth or decay

The base of the exponential function, \(\frac{5}{6}\), is a number between 0 and 1.

Step 3: Conclude whether the function represents growth or decay

Because the base of the exponential function is between 0 and 1, we conclude that this function represents exponential decay.

Key Concepts

Exponential Growth

Exponential growth occurs when a quantity increases by a constant rate or proportion over time. In mathematical functions, we often see it represented as \( f(x) = a imes b^x \), where:

• \( a \) is the initial amount or size of the quantity,
• \( b \) is the base of the exponential function,
• \( x \) is the exponent or the variable representing time or another independent factor.

The key feature of exponential growth is that the base, \( b \), is greater than 1. This means that with each increase in \( x \), the function's value multiplies by a fixed factor greater than 1, leading to a rapid increase.
Real-world examples include population growth, where every generation reproduces more individuals, or compound interest in finance, where interest earnings themselves earn interest, accelerating the growth of an investment. Whenever you see a base greater than 1 in a mathematical function, it typically indicates exponential growth.

Exponential Decay

Exponential decay is the inverse of exponential growth. It occurs when a quantity decreases by a constant rate or proportion over time. Mathematically, it can still be expressed using the formula \( f(x) = a imes b^x \), with the difference being that:

• \( a \) is the initial amount, just like in growth situations,
• \( b \) is the base of the exponential, but it is a number between 0 and 1,
• \( x \) is still the exponent, representing progression over time.

The base \( b \) being between 0 and 1 means that with each increment of \( x \), the outcome is multiplied by a fraction and thus decreases. It's like continually taking smaller and smaller bites out of what's remaining.
Common examples of exponential decay include radioactive decay, where unstable atoms lose their radioactivity at a consistent rate, and depreciation of assets, where the value of an asset reduces consistently over time. Spotting a base less than 1 signifies that the function represents exponential decay, as in the given exercise where \( \frac{5}{6} \) leads to decay.

Mathematical Functions

Mathematical functions are like a machine that takes an input value, applies certain operations, and produces an output value. In the realm of algebra, functions are often expressed in terms like \( f(x) \), where \( f \) indicates the function and \( x \) represents the input variable.

• Functions provide a relationship between the input and the output.
• They can model many real-life situations, helping in forecasting and decision-making.

A crucial aspect of functions is evaluating how they behave as inputs change, which involves examining concepts like growth, decay, maximums, and minimums.
Understanding exponential functions, whether they showcase growth or decay, is a part of this larger picture. These functions are noted for their rapid change in values, unlike linear functions that change at a steady rate. This makes them particularly useful in scenarios where such rapid changes occur naturally, such as economic growth models or understanding how populations evolve over time. Functions, therefore, serve as vital tools in mathematics and science, providing clear and predictive power to various phenomena.