Domain Of An Exponential Function

An exponential function is a office in which the independent variable is an exponent. Exponential functions have the general class y = f (x) = a ^x , where a > 0, a≠1, and x is whatever existent number. The reason a > 0 is that if it is negative, the role is undefined for -ane < x < 1. Restricting a to positive values allows the role to take a domain of all real numbers. In this example, a is chosen the base of operations of the exponential function.

Here is a petty review of exponents:

exponent

a -x = .

a x-y = .

a ten = a y;if and only if;x = y.

Below are pictured functions of the form y = f (x) = a ¹⁰ and y = f (x) = a ^-x . Study them.

Figure %: The graphs of y = ii^x andy = 2^- x .

The domain of exponential functions is all existent numbers. The range is all real numbers greater than zero. The line y = 0 is a horizontal asymptote for all exponential functions. When a > one: every bit x increases, the exponential office increases, and as x decreases, the function decreases. On the other hand, when 0 < a < ane: as 10 increases, the function decreases, and as ten decreases, the function increases.

Exponential functions have special applications when the base of operations is e . eastward is a number. Its decimal approximation is about 2.718281828. It is the limit approached by f (ten) when f (10) = (1 + )¹⁰ and ten increases without bound. Go alee and plug the equation into your computer and check it out. e is sometimes called the natural base of operations, and the function y = f (x) = e ^x is called the natural exponential office.

The natural exponential function is especially useful and relevant when it comes to modeling the behavior of systems whose relative growth rate is abiding. These include populations, bank accounts, and other such situations. Let the growth (or decay) of something be modeled past the function f (x), where ten is a unit of measurement of time. Let its relative growth charge per unit ( ) be the constant one thousand . And then its growth is modeled past the exponential part f (x) = f (0)due east ^kx . Given any ii of the post-obit values: f (0), k , or x , the 3rd can be calculated using this office. In Applications we'll see some useful applications of this role.

Domain Of An Exponential Function,

Source: https://www.sparknotes.com/math/precalc/exponentialandlogarithmicfunctions/section1/

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