Exponential Function Reference

This is the Exponential Function:

f(x) = a^x

a is any value greater than 0

Properties depend on value of "a"

When a=1, the graph is a horizontal line at y=1
Apart from that there are two cases to look at:

a between 0 and 1		a above 1
Example: f(x) = (0.5)x		Example: f(x) = (2)x
For a between 0 and 1 As x increases, f(x) heads to 0 As x decreases, f(x) heads to infinity It is a Strictly Decreasing function (and so is "Injective") It has a Horizontal Asymptote along the x-axis (y=0).		For a above 1: As x increases, f(x) heads to infinity As x decreases, f(x) heads to 0 it is a Strictly Increasing function (and so is "Injective") It has a Horizontal Asymptote along the x-axis (y=0).

Plot the graph here (use the "a" slider)

In General:

It is always greater than 0, and never crosses the x-axis
It always intersects the y-axis at y=1 ... in other words it passes through (0,1)
At x=1, f(x)=a ... in other words it passes through (1,a)
It is an Injective (one-to-one) function

Inverse

So the Exponential Function can be "reversed" by the Logarithmic Function.

The Natural Exponential Function

This is the "Natural" Exponential Function:

f(x) = e^x

Where e is "Eulers Number" = 2.718281828459 (and more ...)

Graph of f(x) = e^x

At the point (1,e) the slope of the line is e and the line is tangent to the curve.