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Section 8.2Inverse Functions
• Consider the following two tables
• Are they both functions?• What are f-1 and g-1?
x f(x)
-2 10
-1 12
0 16
1 20
2 27
3 44
x g(x)
-5 -10
-2 -12
-1 -16
1 -16
2 -12
5 -10
Definition of an Inverse Function• Suppose Q = f(t) is a function with the
property that each value of Q determines exactly one value of t. The f has an inverse function, f -1 and
If a function has an inverse, it is said to be invertible
).(ifonlyandif)(1 tfQtQf
Graphs of Inverses• Consider the function 3)( xxf
• The inverse is 3)( 21 xxf
Plot of the two graphs together
Horizontal Line Test
• A function must be one-to-one in order to have an inverse (that is a function)
• A function is one-to-one if it passes the horizontal line test– A horizontal line may hit a graph in at most one
point
• We can restrict the domain of functions so their inverse exists– For example, if x ≥ 0, then we have an inverse for
3)( 2 xxg
• Let’s talk about the following problems
)100(find1)(Given
)20(find23)(Given
)(find13)(Given
)(find13)(Given
15
13
1
13
kxxxk
hxxh
xgxg
xfxxf
x
x
Property of Inverse Functions• If y = f(x) is an invertible function and y = f -1(x) is
its inverse, then
definedis)(
whichforofvaluesallfor))((
definedis)(
whichforofvaluesallfor))((
1
1
1
xf
xxxff
xf
xxxff
• In your groups try problems 5, 15, and 25