Exponential Functions

Exponential Functions are mathematical functions that are useful for describing relationships, primarily those involving Exponential Growth and Decay. They are expressed algebraically as:

\(f(x) = b^x\)

In this equation, \(b\) represents the Growth or Decay Factor.

An exponential function has a repeating pattern of finite differences. Unlike Linear Functions which change at a constant rate, Exponential Functions change by a common ratio.

Growth and Decay Factor

The Growth/Decay Factor, 𝑏, is used to determine whether the quantity rapidly increases or decreases.

Growth Factor

When \(b>1\), the function has a Growth Factor. This means that it contains a rate of change that increases over time. Initially, the function slowly increases and then increases more sharply over time.

Exponential Function representing Exponential Growth. The function rapidly increases as it moves left to right.

Regarding growth, if a quantity were to double in value each period, \(b=2\). Likewise, if a quantity were to increase by \(5\%\) each period, \(b=1.05\).

Decay Factor

When \(0 < b < 1\), the function has a Decay Factor. This means that it contains a rate of change that decreases over time. Initially, the function decreases sharply and then gradually decreases over time.

Exponential Function representing Exponential Decay. The function rapidly descreases as it moves left to right.

Conversely, regarding decay in relation to half-life, \(b=0.5\). Similarly, if a quantity were to decrease by \(8\%\) each period, \(b=0.92\).

The growth/decay factor, \(b\), is always a positive constant where \(b \ne 1\). Otherwise, it will appear graphically as a straight horizontal line.

Inverse Exponential Functions

The inverse of the standard exponential function can be expressed algebraically as:

\(x = b^y\)

In this equation, \(b\) represents the Growth or Decay factor and y represents the variable. The inverse exponential of a function where \(b>1\) can be graphed in relation to the original function as such:

Graph showing regular Exponential Function against Inverse Exponential Function where b>1.

The inverse exponential of a function where \(0 < b < 1\) can be graphed in relation to the original function as such:

Graph showing regular Exponential Function against Inverse Exponential Function where 0<b<1.

Characteristics of Exponential Functions

The Domain represents a function’s input values. For exponential functions, it is represented as \(x\) as an element of all real numbers, or \(\{x\in\mathbb{R}\}\).

The Range represents a function's output values. For exponential functions, it is represented as \(y\) as an element of all real numbers greater than \(0\), or \(\{y\in\mathbb{R} |y>0\}\).

Horizontal Asymptotes are horizontal lines that a function cannot cross, thereby restricting its domain. Exponential functions contain a horizontal asymptote at \(y=0\).