Exponential Function Reference

This is the general Exponential Function (see below for ex):

f(x) = ax

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

a above 1

exponential function
Example: f(x) = (2)x

For a above 1:

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: Real Numbers

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


ax   is the inverse function of   loga(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) = ex

Where e is "Eulers Number" = 2.718281828459... etc

natural exponential function
Graph of f(x) = ex

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

At any point the slope of ex equals the value of ex :

natural exponential function
when x=0, the value of ex = 1, and slope = 1
when x=1, the value of ex = e, and slope = e

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

natural exponential function