The exponential function in algebra is defined as a mapping where the independent variable appears exclusively in the exponent with a real base greater than zero but not equal to one, forming a subfield of elementary transcendental functions within mathematical analysis. The core theoretical principle establishes that this function maps every element of its domain (the set of all real numbers) uniquely onto its range via continuous growth or decay characterized by specific asymptotic behaviors and inverse logarithmic relationships. This formal structure provides the necessary algebraic foundation for understanding non-polynomial functional equations, specifically distinguishing it from polynomial power functions through properties such as $f(x+y)=f(x)f(y)$ under standard notation conventions where applicable to bases other than $e$.
Introduces exponential functions: the variable in the exponent, growth versus decay, key graph features like the y-intercept and asymptote, and contrasting exponential with linear change.