Domain of Exponential Functions

Introduction to domain of exponential functions:

The exponential function in mathematics is defined as ex, where e is an integer. For example let us denote ex is an exponential function, Let us assume the value of x is zero, x = 0 then the solution is in the form of e0 = 1. Provided e.g. shows general concept of the exponential function. Here we are going to discuss about what is domain and the exponential function together as domain of exponential functions.

Domain of Exponential Functions:

Domain is the set of point or atlest a single point which refers an open connected space. We can say the domain in other words as group of all possible values from an independent variable of a function. The domain and range of the function is lies between (-infinity) to (+infinity).

First we have to start with the fundamental functions of the exponential function of base a,

f (x) = ax , a > 0 and also not equal to 1.

Then the value of the domain functions f is the set of all real numbers.

The range of f is the interval (0 , +infinity).

The given function have a horizontal asymptote which given by y = 0, then the function f has a y intercept at (0, 1).

When the value of the given function f is increased, then the value of a is greater than 1

When the value of the given function f is decreased, then the value of a is lesser than 1.

Example for Domain of Exponential Functions:

Example for domain of exponential functions 1:

Find out the domain of the given function f(x) = ln of (`sqrt (x^2 - 9x + 18)` )

Solution:

Step 1: Given exponential function is f(x) = ln of (`sqrt (x^2 - 9x + 18)` )

Step 2: When we take any value of the 'ln' then it must be a positive value. So:

`sqrt(x^2 - 9x - 18)` > 0

Therefore, `x^2` - 9x - 18 > 0

`x^2 ` - 3x - 6x + 18 > 0

x(x - 6) - 3 (x - 6) > 0

(x - 6)(x - 3) > 0

So either x - 6 and x - 3 are both positive, or x - 6 and x - 3 are both negative. So either x < 6 or x > 3. That's the domain. Having problem with help solve math problem keep reading my upcoming posts, i will try to help you.

Example for domain of exponential functions 2: Determine the domain of the function f(x) = log10 { 1 - log [`x^2` - 4x + 13]}

Solution:

Step 1: The given function is f(x) = log10 { 1 - log [`x^2` - 4x + 13]}

Step 2: Let there are two function present in log. Let we take outer log as nesting log and inner log as nested log.

Step 3: First nesting log is simplified as,

1 - log (`x^2` - 4x + 13)

log (`x^2` - 4x + 13)  < 1

log10 (`x^2` - 4x + 13) < log1010

x2 - 4x + 13 < 10

x2 - 4x + 3 < 0

x - 3x - x + 3 < 0

(x - 3)(x - 1) < 0

x `in` (3, 1)

For nested log function,

`x^2` - 3x + 12 > 0

Here, from the given function squared terms and their co efficient are positive and also domain is > 0. There fore the inequality is true for every real numbers.