Function Exploration

Introduction to Exponential Functions

Exponential functions are mathematical functions that can describe extremely rapid growth or decay. Let's explore the properties of exponential functions through a real-world example.

Virus Spread

Imagine the following scenario: Someone carries a virus and infects 3 other people. Then, each of those people infects 3 more people in the next phase.

Spread Pattern

If we track the number of infected people in each phase:

• Phase 1: $3 = 3^1$ people infected
• Phase 2: $9 = 3^2$ people infected
• Phase 3: $27 = 3^3$ people infected
• Phase 4: $81 = 3^4$ people infected
• Phase 5: $243 = 3^5$ people infected

Mathematical Pattern

From the data above, a clear pattern emerges: the number of people infected in phase $x$ is $3^x$.

If $f(x)$ represents the number of people infected in phase $x$, then:

This is an example of an exponential function.

Spread Visualization

1. Exponential Graph: $f(x) = 3^x$
2. Linear Graph: $f(x) = 3x$
3. Logarithmic Graph: $f(x) = \log(x+1) \cdot 20$

Using the equations above, we can visualize the virus spread through the following graph:

Analysis Questions

1. How many people will be infected in phase 20?

   $$f(20) = 3^{20} = 3,486,784,401$$

2. Which function best represents the virus spread?

   Of the three graphs shown, the exponential graph most accurately depicts this virus spread. This graph shows slow growth initially but becomes very rapid as phases progress.

Properties of Exponential Functions

From the exploration above, we can conclude several properties of the exponential function $f(x) = a^x$ (where $a > 0$ and $a \neq 1$):

1. Rapid Growth/Decay: Function values increase/decrease very rapidly.
2. Domain and Range: Domain is all real numbers, range is all positive numbers.
3. Intercept Point: Always passes through point (0,1) because $a^0 = 1$.
4. Graph Properties:
   • If $a > 1$, the function increases (as in the virus spread case with $a = 3$)
   • If $0 < a < 1$, the function decreases

Applications of Exponential Functions

Exponential functions are used in various fields:

• Population growth
• Compound interest in economics
• Radioactive decay
• Disease spread (as in the example above)

Understanding exponential functions helps us analyze and predict phenomena that exhibit rapid growth or decay.