Inverse Functions
Lesson 8.2
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) }
Example
• Consider an element of an electrical circuit
which increases its resistance as a function
of temperature.
T = Temp
R = Resistance
-20
50
0
150
20
250
40
350
R = f(T)
Example
• We could also take the view that we wish to
determine T, temperature as a function of R,
resistance.
R = Resistance
T = Temp
50
-20
150
0
250
20
350
40
T = g(R)
Now we would
say that g(R) and
f(T) are inverse
functions
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
x 1 
1
x
Does This Have An Inverse?
• Given the function at the right
x
Y
1
5
2
9
4
6
7
5
 Can it have an inverse?
 Why or Why Not?
NO … when we reverse
the ordered pairs, the
result is Not a function.
Finding the Inverse
Given f ( x)  2 x  7
then y  2 x  7
y 7
solve for x
x
2
y 7
1
f  y 
2
Try
x2
y
x2
Composition of Inverse Functions
•
•
•
•
Consider f ( x)  x and
f(3) = 27 and f -1(27) = 3
Thus, f(f -1(27)) = 27
and f -1(f(3)) = 3
3
1
f ( x)  x
3
• In general f(f -1(n)) = n and f -1(f(n)) = n
(assuming both f and f -1 are defined for n)
Graphs of Inverses
f ( x)  x 3
f 1 ( x)  3 x
• Again, consider
• Set your calculator for the functions shown
and
Dotted style
• Use Standard Zoom
•Then use Square Zoom
Graphs of Inverses
• Note the two graphs are symmetric
about the line y = x
Investigating Inverse Functions
• Consider
f ( x)  2 3 x  4
3
x
g ( x)   4
8
• Demonstrate that these are inverse functions
• What happens with f(g(x))?
Define these
functions on your
• What happens with g(f(x))?
calculator and try
them out
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
Domain and Range
• Consider the function h(x) = x2 - 9
• Determine the inverse function
• Problem => f -1(x) is not a function
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)
Note ... the curve is not the
path of the pumpkin.
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.
Assignment
• Lesson 8.2
• Page 370
• Exercises 1 – 49 EOO