Exponential Function Family

An Exponential Function Family is a function family of the form : [math]\displaystyle{ f(x) = b^x \, }[/math] in]] that ...

• Context:
  • It can be instantiated in an Exponential Function Instance.
  • …
• Example(s):
  • a Gaussian Function Family.
• See: e (Mathematical Constant), Graph of a Function, Asymptote.

---

---

References

2016

• (Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/exponential_function Retrieved:2016-8-14.
  • In mathematics, an exponential function is a function of the form : [math]\displaystyle{ f(x) = b^x \, }[/math] in which the input variable x occurs as an exponent. A function of the form f(x) b^{x + c} is also considered an exponential function, and a function of the form f(x) a·b^x can be re-written as f(x) b^{x + c} by the use of logarithms and so is an exponential function.

    Exponential functions are uniquely characterized by the fact that the growth rate of such a function is directly proportional to the value of the function. This proportionality can be expressed by saying : [math]\displaystyle{ \frac {d}{dx} b^x = \frac{d}{dx} \exp(x \ln b) }[/math] :: [math]\displaystyle{ = \exp(x \ln b) \cdot \frac{d}{dx}(x \ln b) }[/math] :: [math]\displaystyle{ = b^x \ln b }[/math] where ln b is a constant, and a constant is a quantity that does not change as the variable x changes.

    For just one base b this constant factor is equal to 1, and that base is the number e ≈ 2.71828...: : [math]\displaystyle{ \frac d {dx} e^x = e^x \times 1 }[/math] This equality makes it possible to reduce some questions in mathematical analysis of exponential functions to the analysis of this one exponential function, conventionally called the "natural exponential function" and denoted by : [math]\displaystyle{ x \mapsto \exp(x). \, }[/math] The exponential function models a relationship in which a constant change in the independent variable gives the same proportional change (i.e. percentage increase or decrease) in the dependent variable. The function is often written as exp(x), especially when it is impractical to write the independent variable as a superscript. The exponential function is widely used in physics, chemistry, engineering, mathematical biology, economics and mathematics. The graph of y = e^x is upward-sloping, and increases faster as x increases. The graph always lies above the x-axis but can get arbitrarily close to it for negative x; thus, the x-axis is a horizontal asymptote. The slope of the tangent to the graph at each point is equal to its y-coordinate at that point. The inverse function is the natural logarithm ln(x); because of this, some old texts refer to the exponential function as the antilogarithm.

    In general, the variable x can be any real or complex number or even an entirely different kind of mathematical object; see the formal definition below.

---