Names
Inverse Functions
 

Tom Linton, http://www.central.edu/homepages/lintont

The plot below shows the balance, B(t) (in dollars) of Tom's savings account in year t, where t is measured in years with t = 0 corresponding to the year 1990. Use the graph to answer the following questions
 
 

[Maple Plot]

  1. When did Tom have $650 in his account?

    2.  

     
     
     
     
     
     
     
     
     
     
     
     

  2. When did Tom have $800 in his account?

    3.  

     
     
     
     
     
     
     
     
     
     
     
     

  3. Does any account balance occur more than once?


The first two questions above are similar to questions we ask all the time:

What input value x gives a specific output value (like 650 or 800)?
The third question is the test for whether such questions have a single answer. If some account balance, say $700, occurred more than once, the question "When did Tom have $700 in his account?" would have multiple correct answers. The relationship which takes a balance of Tom's account, i.e. a value in the range of B(t), and produces the value of t when the account obtained that balance, would NOT be a function. It would still be a legal relationship and quite valuable at that, but it wouldn't be a function. Questions like the first two above occur often enough that special terminology has been created to easily express such questions. When we reverse the roles of input and output (by giving an output value and asking what input produces it), we are asking for values of the inverse. Inverses exist for all functions, but usually the inverse is not a function. The function B(t) plotted above has an inverse which is a function. The reason is that for all balances in the range of B, there is only one time when the account takes on that balance. Put another way, the graph of B(t) passes the horizontal line test.
The Horizontal Line Test
If every horizontal line intersects the graph of f(x) in either zero or one place(s), then f(x) passes the horizontal line test. In these situations, the inverse of f(x) is a function. All monotone functions (those which are always increasing and those which are always decreasing) will pass the horizontal line test.

Look at plots of the following functions and decide which pass the horizontal line test.

  1. f(x) = 2 / (e^x + 5)
  2. g(t) = 1 + ln(t)
  3. h(z) = z^2 + 2^z
Whenever f(x) passes the horizontal line test, the function f(x) has an inverse which is also a function. The term inverse comes from the fact that f(x) and its inverse undo one another. The symbol f -1(x) is used for the inverse of f(x) (whenever the inverse is a function). It does NOT mean 1 / f(x) however. If f(8) = 3, then f -1(3) = 8. The inverse takes an output from f and produces the input value of x needed to obtain that output. The question "what is f -1(12)?" is asking what value of x is needed as input to f(x) so that the output is 12. The first question about Tom's account balance can be restated in inverse terms as
What is B-1(650)?
Restate the second question about Tom's account balances in terms of inverse functions.
 
 
 

Given two formulas for functions f(x) and g(x), they are inverses whenever f( g(x) ) = x and g( f(x) ) = x (first one then the other, does nothing). This algebraic test verifies that whatever f does to x, g undoes to f(x) and whatever g does to x, f undoes to g(x). To run this test, you simply compute a formula for f( g(x) ) and for g( f(x) ) and try to simplify each to just x. If both simplify to x, they are inverses. If one fails to simplify to just x, they are NOT inverses.
Which of the pairs below are inverses?

  1. f(x) = 4 / (2x + 3) and g(x) = 2 / x - 3 / 2

    2.  

     
     
     
     
     
     
     
     
     
     
     
     
     

  2. f(x) = 2 / (e^x + 5) and g(x) = ln( 2 / x  - 5)

    3.  

     
     
     
     
     
     
     
     
     
     
     
     
     
     

  3. f(x) = 5 / (2x - 3) and g(x) = (5x - 3) / 2

 
 

Graphs of f(x) and f -1(x) are reflections of one another about the line y = x. Many times, the scales needed to see both f and its inverse make it difficult to "see" this reflective property. Here is one such picture where the scales are fine. The line y = x is plotted as well (dashed). f(x) is thick and f -1(x) is thin.

Draw a graph of an increasing function f(x) and then draw its inverse (on the same axes).