Function Definition

Exponential Function

An exponential function is a function expressed in the form:

with the conditions:

• $a$ is the base number, where $a > 0$ and $a \neq 1$
• $n$ is a non-zero real number
• $x$ is any real number

Exponential functions have a special characteristic where the variable $x$ is in the exponent position. This is what distinguishes exponential functions from ordinary algebraic functions. In exponential functions, small changes in the value of $x$ can result in very large changes in the function's output.

Properties of Exponential Functions

The exponential function $f(x) = a^x$ (for $n = 1$) has several important properties:

1. The domain of the function is all real numbers ($\mathbb{R}$)
2. The range of the function is all positive numbers ($\mathbb{R}^+$)
3. It intersects the vertical axis at point $(0, 1)$ because $a^0 = 1$
4. The function is always positive for all values of $x$ because $a > 0$
5. If $a > 1$, the function increases (monotonically increasing)
6. If $0 < a < 1$, the function decreases (monotonically decreasing)

Special Cases of Exponential Functions

When the Base Is One

If $a = 1$, then:

The value of $1^x$ is always $1$ for any value of $x$. As a result, the function becomes a constant function $f(x) = n$, no longer an exponential function. Its graph will be a horizontal line intersecting the vertical axis at point $(0, n)$.

When the Base Is Zero

If $a = 0$, then:

• For $x > 0$, the value of $0^x = 0$ so $f(x) = 0$
• For $x = 0$, the value of $0^0$ is undefined
• For $x < 0$, the value of $0^x$ is undefined

This function is no longer an exponential function but rather constant at $f(x) = 0$ for $x > 0$. Then, because $0^0$ and $0^x$ for $x < 0$ are undefined, this function does not meet the definition of an exponential function.

Examples of Exponential Functions

Here are some examples of exponential functions:

1. $f(x) = 4^x$

   This function has base number $a = 4$ and $n = 1$. Since $a > 1$, this function is monotonically increasing. The function value will get larger as $x$ increases. For example, $f(0) = 4^0 = 1$, $f(1) = 4^1 = 4$, $f(2) = 4^2 = 16$.

2. $f(x) = 3^{x+1}$

   This function can be rewritten as $f(x) = 3 \times 3^x$ with base number $a = 3$ and $n = 3$. The graph of this function is also monotonically increasing, and the function value will get larger as $x$ increases. For example, $f(0) = 3^{0+1} = 3^1 = 3$, $f(1) = 3^{1+1} = 3^2 = 9$.

3. $f(x) = 5^{2x-1}$

   This function has base number $a = 5$ with exponent $2x-1$. The function value will change more rapidly because the coefficient of $x$ is $2$. For example, $f(0) = 5^{2(0)-1} = 5^{-1} = \frac{1}{5}$, $f(1) = 5^{2(1)-1} = 5^1 = 5$.

4. $f(x) = 0.5^x$

   This function has base number $a = 0.5$ where $0 < a < 1$. This function is monotonically decreasing. The function value will get smaller as $x$ increases. For example, $f(0) = 0.5^0 = 1$, $f(1) = 0.5^1 = 0.5$, $f(2) = 0.5^2 = 0.25$.

Applications of Exponential Functions

Exponential functions are widely used in everyday life and various fields:

1. Population Growth: The number of bacteria reproducing can be modeled with an exponential function $P(t) = P_0 \times 2^{t/n}$ where $P_0$ is the initial number, $t$ is time, and $n$ is the time required for the population to double.

2. Compound Interest: If someone saves money with compound interest, the amount of savings after $t \text{ years}$ can be calculated with $A(t) = P \times (1 + r)^t$ where $P$ is the initial principal and $r$ is the interest rate.

3. Radioactive Decay: The amount of radioactive substance remaining after $t \text{ years}$ can be calculated with $A(t) = A_0 \times 0.5^{t/h}$ where $A_0$ is the initial amount and $h$ is the half-life.

4. Virus Spread: The spread of disease in a population often follows an exponential model in the early phase.