Exponential Function Reference

This is the Exponential Function:

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, +∞)

Inverse

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 (and more ...)

natural exponential function
Graph of f(x) = ex

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