Most of your experience thus far has likely been with functions where the
base is a variable, and the exponent is a constant, like these:
Functions with a variable base and a constant exponent are called power functions.
Switch the role of variable and constant and you get an entirely new family of functions,
called exponential functions, which are the subject of this section.
For example, these are all exponential functions:
The precise definition follows:
An exponential function is a function of the form $\,y=b^x\,$,
where $\,b\,$ is a positive number not equal to $\,1\,$,
and $\,x\,$ is any real number.
Thus, exponential functions have a constant base; the variable is in the exponent.
The number $\,b\,$ is called the base of the exponential function.
The most important exponential function is when the base is the irrational number $\,\text{e}\,$.
(Note: $\text{e}\approx 2.71828\,$)
In this case, the function is also written as $\ \exp(x)\,$,
and is called the natural exponential function.
If you hear the phrase ‘the exponential function’ (meaning only one)
then the function being referred to is $\,y={\text{e}}^x=\exp(x)\,$.
Some mathematicians (this author included) tend to call more general functions
with a constant base and variable in the exponent exponential functions;
say, $\ y={\text{e}}^{2x3}\ $ or $\ y=7\cdot 3^{5x}\,$.
However, for the purposes of this section,
the phrase exponential function refers
only to functions of the form
$\,y=b^x\,$.
Why must the base of an exponential function be positive?
For negative numbers, there are problems for many values of $\,x\,$;
for example, $\,(4)^{1/2} = \sqrt{4}\,$ is not a real number.
Why can't the base be $\,1\,$ or $\,0\,$?
The base can't be $\,1\,$, because $\,1^x \ \overset{\text{always}}{=}\ 1\,$, so the function
$\,1^x\,$ is a constant function.
There is a similar problem with zero as the base.
There are two basic shapes to the graphs of exponential functions,
depending on whether the base is greater than $\,1\,$, or between $\,0\,$ and $\,1\,$:
$\,y=b^x\,$ for $\,b\gt 1\,$  $\,y=b^x\,$ for $\,0\lt b\lt 1\,$ 
increasing functions;
also called growth functions 
decreasing functions; also called decay functions 
PROPERTIES OF THE GRAPH for $\,b\gt 1\,$
An increasing function has the following property:
The following are equivalent for a function $\,f(x)=b^x\,$:
For increasing exponential functions:
It is important to note that increasing exponential functions increase VERY quickly.
Even more impressive is this fact: 
$y = b^x\,$ for $\,b \gt 1$
increasing functions as $\,x\rightarrow \infty \,$, $\,y\rightarrow \infty \,$ as $\,x\rightarrow \infty \,$, $\,y\rightarrow 0\,$ 
PROPERTIES OF THE GRAPH for $\,0\lt b\lt 1\,$
A decreasing function has the following property: The following are equivalent for a function $\,f(x)=b^x\,$:
For decreasing exponential functions:

$y = b^x\,$ for $\,0 \lt b \lt 1$
decreasing functions as $\,x\rightarrow \infty \,$, $\,y\rightarrow 0 \,$ as $\,x\rightarrow \infty \,$, $\,y\rightarrow \infty\,$ 
Let $\,f(x)=b^x\,$, where $\,b\,$ is a positive number not equal to $\,1\,$.
For all (allowable) bases $\,b\,$, exponential functions share the following properties:
THE DOMAIN IS THE SET OF ALL REAL NUMBERS:
$\text{dom}(f) = \mathbb{R}$
If the graph of an exponential function is ‘collapsed’ into the $\,x$axis, Exponential functions know how to act on all real number inputs.
For basic information on the domain and range of a function, 

THE RANGE IS THE SET OF ALL POSITIVE REAL NUMBERS:
$\text{ran}(f)=(0,\infty )$
If the graph of an exponential function is ‘collapsed’ into the $\,y$axis, Outputs from exponential functions are always positive.
Having trouble understanding the expression ‘$\,(0,\infty)\,$’? 

THE GRAPH CROSSES THE $\,y\,$AXIS AT $\,y=1\,$
For allowable values of $\,b\,$,
$\,b^0 \ \overset{\text{always}}{\ \ =\ \ }\ 1\,$. 

Imagine a vertical line sweeping through a graph, checking each allowable $\,x$value:
if it never hits the graph at more than one point, then the graph is said to pass the vertical line test.
All functions pass the vertical line test, since the function property is that
each input has exactly one output.
passes the vertical line test: each $\,x$value has only one $\,y$value all functions pass the vertical line test 
fails the vertical line test: there exists an $\,x$value that has more than one $\,y$value 
Imagine a horizontal line sweeping through a graph, checking each allowable $\,y$value:
if it never hits the graph at more than one point, then the graph is said to pass the horizontal line test.
Some functions pass the horizontal line test, and some do not.
passes the horizontal line test: each $\,y$value has only one $\,x$value all exponential functions pass the horizontal line test 
fails the horizontal line test: there exists a $\,y$value that has more than one $\,x$value some functions fail the horizontal line test 
Thus, exponential functions have a wonderful property:
each input has exactly one output (passes the vertical line test),
and
each output has exactly one input (passes the horizontal line test).
For such functions, you can think of the inputs/outputs as being connected with strings:
pick up any input, and follow its ‘string’ to the unique corresponding output;
pick up any output, and follow its ‘string’ to the unique corresponding input.
That is, there is a onetoone correspondence between the inputs and outputs.
Functions with this property are called onetoone functions.
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
