# Introduction to Exponential Functions (Part 2)

(This page is Part 2. Click here for Part 1.)

## Properties that All Exponential Functions Share

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: $\,\text{dom}(f)=\mathbb{R}\,$
- The range is the set of all positive real numbers: $\,\text{ran}(f)=(0,\infty)\,$
- The graph crosses the $y$-axis at $\,y = 1$
- The graph passes both the vertical and horizontal line tests

### 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, sending each point on the graph to its $x$-value, then the entire $x$-axis will be hit.

Exponential functions know how to act on all real number inputs.

For basic information on the domain and range of a function, you may want to review: 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, sending each point on the graph to its $y$-value, then all positive $y$-values will be hit.

Outputs from exponential functions are always positive.

Having trouble understanding the expression ‘$\,(0,\infty)\,$’ ? Then, you may want to review Interval and List Notation.

### The Graph Crosses the $y$-Axis at $\,y=1\,$

For allowable values of $\,b\,,$ $\,b^0 \ \overset{\text{always}}{\ \ =\ \ }\ 1\,.$ So, when the input is $\,0\,$ to the function $\,y=b^x\,,$ the output is $\,1\,.$ Thus, the point $\,(0,1)\,$ lies on the graph of every exponential function.

### The Graph Passes Both the Vertical and Horizontal Line Tests

Vertical Line Test:
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

Horizontal Line Test:
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
*one-to-one correspondence*
between the inputs and outputs.
Functions with this property
are called *one-to-one functions*.