A function is defined on two sets, one of which is the input set, while another is the output set. A relation between input and output sets is called a function if they are related in such a way that every element of input set is connected with one and only one output, but an output can be produced from more than one inputs though. Generally, the functions are represented by letters "f" and "g" by the equation y = f(x) or y = g(x), where x denotes independent variable and y represents dependent variable. There are many different types of functions in mathematics. The exponential functions are ones that are most common.

The definition of a general exponential function is that these are functions of the type y = $d^{x}$, where d > 0 and also 'd' is not equal to 1. In this case 'd' is known as the base and 'x' is the independent variable that can take any real values.

Consider d = 0. In this case y = $0^{x}$ = 0 and when d = 1 then y = $1^{x}$ = 1 which are not exponential functions. This is why we don’t take value of d as zero or one. Also, d cannot take negative values this is because at times we can have x = $\frac{1}{n}$ and in such case we cannot find nth root of any negative number in terms of real values. So, we can see that the restrictions that are applied on base ‘d’ are completely valid.

Formula

y = g (x) = $d^{x}$, where d > 0 and d $\neq$ 1.

This is the standard equation of a function of exponential type where x is any variable taking all real values.
Natural Exponential Function:

The natural function in exponential form is given by:

z = f(y) = $e^{y}$

Where, 'e' is the Euler’s number which is a constant whose value is 2.71828182…

Properties

The properties of exponential functions are discussed below:

a) 
For any given exponential function g (0) = 1, that is, the graph of such function will always be having the point (0, 1) no matter what is the value of d.

b) The domain of exponential function is the set of all real numbers.

c) The range of an exponential function is the set of all positive real numbers excluding zero.

d) X-axis serves as an asymptote to the graph of any given exponential function. 
 
e) For all values of d, $d^{x}$ > 0 that is $d^{x}$ can never be equal to zero.

f) If d lies between 0 & 1 (0 < d < 1) then the graph of the function in this case as we go from left to right will always decrease.

g) If d is greater than 1 (d > 1), then the graph of the function in this case as we go from left to right will always be increasing.

h) Suppose we are given that $d^{x}$ equals $d^{z}$. This will imply that x and z are taking equal values. Mathematically,
$d^{x} = d^{z} \Rightarrow x = z$

Examples

Have a look at the following examples related to exponential functions:

Example 1: Evaluate $(\frac{25}{16})^{\frac{-5}{2}}$.

Solution: $(\frac{25}{16})^{\frac{-5}{2}}$

= $(\frac{5^{2}}{4^{2}})^{\frac{-5}{2}}$

= $(\frac{4^{2}}{5^{2}})^{\frac{5}{2}}$

= $[(\frac{4}{5})^{2}]^{\frac{5}{2}}$

= $(\frac{4}{5})^{2 \times \frac{5}{2}}$

= $(\frac{4}{5})^{5}$

$\frac{1024}{3125}$

Example 2: Given th
at the population of a certain town in the year 1991 is 20,000. Also it is known that there is an annual growth in population by 1.5%.

a) Find the factor of growth.

b) Make a general equation to find the growth in coming years.

Solution: The population in 1991 is 20,000

Population is increasing at the rate of 1.5% that is 0.015 times annually.

This implies population in 1992 will be

20,000 + 0.015 x 20, 000

= 20,000 (1 + 0.015)

= 20,000 (1.015)

a) Hence, the growth factor = 1.015.

b) Now, we have growth factor = 1.015. Let the time variable be ‘t’ in years and the base population taken as 20,000
Then, the equation for calculating growth in coming years can be written as:

P (t) = 20,000 (1. 015)$^{t}$