Exponential Function Domain And Range

An exponential function is a office in which the independent variable is an exponent. Exponential functions have the general form y = f (10) = a ¹⁰ , where a > 0, a≠1, and x is whatsoever real number. The reason a > 0 is that if it is negative, the function is undefined for -1 < 10 < 1. Restricting a to positive values allows the function to have a domain of all existent numbers. In this instance, a is called the base of the exponential function.

Hither is a little review of exponents:

exponent

a ^-10 =

a ^10-y =

a ^x = a ^y;if and but if;x = y.

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

Figure %: The graphs of y = 2^x *andy* = 2^- 10 .

The domain of exponential functions is all real numbers. The range is all real numbers greater than nil. The line y = 0 is a horizontal asymptote for all exponential functions. When a > 1: as x increases, the exponential role increases, and every bit x decreases, the function decreases. On the other mitt, when 0 < a < i: equally x increases, the function decreases, and every bit x decreases, the function increases.

Exponential functions accept special applications when the base is east . due east is a number. Its decimal approximation is about two.718281828. It is the limit approached by f (x) when f (x) = (ane + )^x and 10 increases without leap. Go ahead and plug the equation into your calculator and cheque it out. eastward is sometimes called the natural base of operations, and the function y = f (ten) = due east ^x is called the natural exponential function.

The natural exponential role is specially useful and relevant when it comes to modeling the behavior of systems whose relative growth rate is constant. These include populations, banking concern accounts, and other such situations. Let the growth (or decay) of something be modeled by the role f (x), where x is a unit of time. Let its relative growth rate ( ) be the constant k . And so its growth is modeled by the exponential function f (x) = f (0)eastward ^kx . Given any two of the post-obit values: f (0), k , or ten , the third can be calculated using this part. In Applications we'll meet some useful applications of this function.

Exponential Function Domain And Range,

Source: https://www.sparknotes.com/math/precalc/exponentialandlogarithmicfunctions/section1/#:~:text=The%20domain%20of%20exponential%20functions,real%20numbers%20greater%20than%20zero.

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