Exponential functioms

An exponential function is a mathematical function of the form $f(x) = a \cdot b^x$, where $b$ is a positive real number not equal to 1, and $a$ is a non-zero real number. In this function:

• $b$ is called the base of the exponential function.
• $x$ is the exponent and represents the independent variable.
• $a$ is the initial value or $y$-intercept (when $x=0$, $f(0) = a \cdot b^0 = a \cdot 1 = a$).

Exponential functions are characterized by a constant ratio of consecutive $y$-values for equal intervals of $x$-values, leading to rapid growth or decay.

For example, $f(x) = 2^x$ is an exponential function where the base is 2.
If $x=0$, $f(0) = 2^0 = 1$.
If $x=1$, $f(1) = 2^1 = 2$.
If $x=2$, $f(2) = 2^2 = 4$.