A Double Exponential Equation

Since and , there are points on the graphs of and where . These graphs are the special cases of where and . All points with can be found as intersections of the graph with the lines with slope . In this case, parametric equations in terms of have simple formulas.
The graph of is black. The graph of interest, where , is blue for and red for , and is the graph of a function . The intersection points with and , for , and corresponding points on , are plotted.
It is interesting to see that when is varied between 0 and 2, the graph of bows from concave up to concave down, and appears to be a line segment from to for some . The graphs of and are shown to help you decide whether the graph of for this really is straight. The special satisfies .
The case is especially interesting because then the equation is equivalent to , which has a solution and . (The slider for can take values from -2 to 5.)


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


Assume . Conveniently, iff iff iff . Thus, is on the graph of where iff and , where .
Let and . Then and , and are parametric equations for the graph of . Since the graph of is symmetric with respect to , and .
Parametric equations for the graphs of and are obtained by differentiating the identity . They are , , and , .
Where does the graph of meet the line ? Two ways to determine it are 1) observe it is the point such that , where , and 2) compute using L'Hospital's rule. You will find .
What are the domain and range of ? It is an interesting exercise in L'Hospital's rule to determine that and when and and when . By symmetry, the domain and range of are when and when .
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2017 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+