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Exponential Function Reference

This is the general Exponential Function (see below for e^x):

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

exponential function
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)

a above 1

exponential function
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)

Try different values of a here:

../algebra/images/function-graph.js?fn0=a^x&xmin=-3.711&xmax=4.028&ymin=-0.7247&ymax=3.675&vara=0.5|0|5

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

a^x is the inverse function of log_a(x) (the Logarithmic Function)

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 so on

natural exponential function
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 :

natural exponential function
when x=0, the value of e^x = 1, and slope = 1
when x=1, the value of e^x = e, and slope = e
and so on...

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

natural exponential function

Exponents and Logarithms Common Functions Reference Algebra Index