Exponential Function Reference

This is the general Exponential Function (see below for e 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

Example: f(x) = (0.5) 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).

Example: f(x) = (2) x

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

Its Domain is the Real Numbers:

Its Range is the Positive Real Numbers: (0, +∞)

Inverse

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

The Natural Exponential Function

This is the "Natural" Exponential Function:

Where e is "Eulers Number" = 2.718281828459. etc

Graph of f(x) = e x

The value e is important because it creates these useful properties:

At any point the slope of e x equals the value of e x :

when x=0, the value of e x = 1, and slope = 1
when x=1, the value of e x = e, and slope = e
etc.

The area up to any x-value is also equal to e x :