The function $f(x) = 100(0.7)^x$ is a classic example of an exponential decay function. Let’s break it down to understand it better.
Understanding the Components
Exponential Function
An exponential function is a mathematical expression in which a constant base is raised to a variable exponent. In general, it can be written as $f(x) = a b^x$, where:
- $a$ is the initial value or coefficient
- $b$ is the base of the exponential
- $x$ is the exponent
In our case, $a = 100$, $b = 0.7$, and $x$ is the variable.
Initial Value
The initial value $a$ is 100. This is the value of the function when $x = 0$. So $f(0) = 100(0.7)^0 = 100$. This means the function starts at 100.
Base of the Exponential
The base $b$ is 0.7. Since $0 < b < 1$, this indicates that the function is decreasing, which is characteristic of exponential decay.
Exponent
The exponent $x$ is the variable that changes. As $x$ increases, the value of $f(x)$ decreases because the base is less than 1.
Graphing the Function
To visualize this function, let’s consider a few values of $x$:
- When $x = 0$, $f(0) = 100(0.7)^0 = 100$
- When $x = 1$, $f(1) = 100(0.7)^1 = 70$
- When $x = 2$, $f(2) = 100(0.7)^2 = 49$
- When $x = 3$, $f(3) = 100(0.7)^3 = 34.3$
As you can see, the value of $f(x)$ decreases rapidly as $x$ increases.
Real-World Applications
Exponential decay functions like $f(x) = 100(0.7)^x$ are used in various fields such as:
Radioactive Decay
In physics, the amount of a radioactive substance decreases exponentially over time.
Depreciation
In finance, the value of an asset may decrease exponentially over time.
Population Decline
In biology, the population of a species may decrease exponentially due to various factors.
Conclusion
The function $f(x) = 100(0.7)^x$ is an exponential decay function where the value decreases as $x$ increases. Understanding this function helps in various real-world applications such as radioactive decay, asset depreciation, and population decline.