Exponential Functions

Exponential Functions
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Definition:
The exponential function with base b is defined by

Let's examine the function

Check out the table
and the graph.

x	f (x) = y
-2
-1
0
1
2
3

Most exponential graphs resemble this same shape.
This graph is very, very small on its left side and is extremely close to the x-axis. As the graph progresses to the right, it starts to grow faster and faster and shoots off the top of the graph very quickly, as seen at the right.

In a straight line, the "rate of change" remains the same across the graph. In these graphs, the "rate of change" increases or decreases across the graphs.

Characteristics:
Such exponential graphs of the form f (x) = b^x have certain characteristics in common:

Exponential functions are one-to-one functions.

• graph crosses the y-axis at (0,1)

• when b > 1, the graph increases

• when 0 < b < 1, the graph decreases

• the domain is all real numbers

• the range is all positive real numbers (never zero)

• graph passes the vertical line test - it is a function

• graph passes the horizontal line test - its inverse is also a function.

• graph is asymptotic to the x-axis - gets very, very close to the x-axis but does not touch it or cross it.

Natural Exponential Function:

The function defined by f (x) = e^x is called the natural exponential function.

(e is an irrational number, approximately 2.71828183, named after the 18th century Swiss mathematician, Leonhard Euler .)

Notice how the characteristics of this graph are similar to those seen above.

This function is simply a "version" of
where b >1.

Inverse of f (x) = e^x:

Since f (x) = e^x is a one-to-one function, we
know that its inverse will also be a function.

But what is the equation of the inverse of
f (x) = e^x ?
To solve for an inverse algebraically:

• set the equation equal to y

• swap the x and y

• solve for y by rewriting in log form

• log base e is called the natural log, ln x.

The inverses for other exponential functions
are found in this same manner.

Graphing the inverse of the natural exponential function by reflecting it over the identity function,
y = x, shows the natural logarithmic function,
y = ln x.