Inverse Functions

1
Inverse Functions

• Lesson 8.2

2
Definition

• A function is a set of ordered pairs with no two
  first elements alike.
• f(x) (x,y) (3, 2), (1, 4), (7, 6), (9,12)
• But ... what if we reverse the order of the
  pairs?
• This is also a function ... it is the inverse
  function
• f -1(x) (x,y) (2, 3), (4, 1), (6, 7), (12,
  9)

3
Example

• Consider an element of an electrical circuit
  which increases its resistance as a function of
  temperature.

R f(T)
4
Example

• We could also take the view that we wish to
  determine T, temperature as a function of R,
  resistance.

T g(R)
Now we would say that g(R) and f(T) are inverse
functions
5
Terminology

• If R f(T) ... resistance is a function of
  temperature,
• Then T f-1(R) ... temperature is the inverse
  function of resistance.
• f-1(R) is read "f-inverse of R
• is not an exponent
• it does not mean reciprocal

6
Does This Have An Inverse?

• Given the function at the right
• Can it have an inverse?
• Why or Why Not?

NO when we reverse the ordered pairs, the
result is Not a function.
7
Finding the Inverse
8
Composition of Inverse Functions

• Consider
• f(3) 27   and   f -1(27) 3
• Thus, f(f -1(27)) 27
• and f -1(f(3)) 3
• In general   f(f -1(n)) n   and f -1(f(n))
  n(assuming both f and f -1 are defined for n)

9
Graphs of Inverses

• Again, consider
• Set your calculator for the functions shown

Dotted style

• Use Standard Zoom
• Then use Square Zoom

10
Graphs of Inverses

• Note the two graphs are symmetric about the line
  y x

11
Investigating Inverse Functions

• Consider
• Demonstrate that these are inverse functions
• What happens with   f(g(x))?
• What happens with  g(f(x))?

Define these functions on your calculator and try
them out
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Domain and Range

• The domain of f is the range of f -1
• The range of f is the domain of f -1
• Thus ... we may be required to restrict the
  domain of f so that f -1 is a function

13
Domain and Range

• Consider the function    h(x) x2 - 9
• Determine the inverse function
• Problem gt  f -1(x) is not a function

14
Inverse Pumpkins

• In a recent pumpkin launching contest, one
  launcher misfired so that the pumpkin went
  straight up into the air (!!) and came back down
  to land on the launch personnel!  
• Below is the graph of the height of the pumpkin
  as a function of time h(t) 

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Inverse Pumpkins

• What is the "hang time" of the launch?
• Restrict the domain of h(t) so that it has an
  inverse that is a function
• Graph the inverse of this function
• Change the story to go with your new graph. 
  Explain in your story why it makes sense that the
  inverse is a function.

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Assignment

• Lesson 8.2
• Page 370
• Exercises 1 49 EOO