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An exponential function has the general form f(x) = A₀ · bˣ, where A₀ is the initial value, b is a positive base not equal to one, and x is the exponent. The base determines whether the function models growth (b > 1) or decay (0 < b < 1). This structure allows exponential functions to describe processes where change is proportional to the current amount.

When the base is greater than one, the exponential function models growth, with values increasing over time. When the base is between zero and one, it models decay, with values decreasing. For example, a drug with 75% retention per hour uses base 0.75, creating a decay curve that falls rapidly at first then gradually flattens toward zero.

For an exponential decay function modeling real-world quantities like drug concentration, the domain is non-negative real numbers since time cannot be negative. The range includes only positive values because the function approaches but never reaches zero, maintaining a horizontal asymptote at zero.

The One-to-One Property states that if bˣ = bʸ, then x = y. This property allows you to solve exponential equations by comparing exponents when bases are equal. It forms the foundation for solving exponential equations with logarithms, enabling algebraic manipulation of exponential expressions.

Exponential functions model dynamic processes where change occurs at a rate proportional to the current amount. Examples include bacterial proliferation, radioactive substance decay, drug concentration in the bloodstream, and asset depreciation. These applications demonstrate how exponential models describe growth and decay across biological, chemical, and economic systems.

Transformations such as translations, reflections, and stretches alter the graphical representation of exponential functions but preserve their fundamental shape and asymptotic behavior. The curve maintains its characteristic rapid initial change followed by gradual flattening, regardless of how the function is shifted, reflected, or scaled.

The base must be positive to ensure the function produces real outputs for all real exponents. If the base were negative, non-integer exponents would yield complex numbers. The base cannot equal one because 1ˣ always equals 1, creating a constant function rather than a true exponential function with growth or decay behavior.