4.2 Inverse Functions
Proving that talent isn’t all you need to become the
next big thing, which formerly unknown teen has hit it
big, with what some are calling the “worst thing I have
ever heard”, with her song and video entitled “Friday?”
One-to-One
A function is one-to-one if for each y value there is only one
corresponding x value. Otherwise, it is many-to-one.
many-to-one
one-to-one
Horizontal Line Test: If a horizontal line intersects the graph of a
function f in only one point, then f is one-to-one.
Inverses
Let f denote a one-to-one function. The inverse is denoted f -1(x)
1
1
and
f ( f ( x))  f ( f ( x))  x
Verify that f and g are inverses of one another.
f ( x)  2 x  6
1
g ( x)  x  3
2
Graphs of Inverse Functions
The graph of a function f and its inverse f -1 are symmetric with
respect to the line y = x.
Finding Inverses
x 9
Find the inverse of f ( x) 
where x  0 .
2
2
Finding Inverses
2x  3
Find the inverse of f ( x) 
where x  4.
x4
Finding Inverses
Domain
Range
Domain
Range
x2  9
f ( x) 
,x0
2
f 1 ( x)  2 x  9
2x  3
f ( x) 
x4
4x  3
f 1 ( x ) 
x2
The domain of f is equal to the range of f -1 and vice versa.
Using Inverses to Find Range
x 3
Find the inverse of f ( x) 
, x  0 . Then find the
2
3x
range of f using f -1.
2
4.2 Inverse Functions
Homework:
pgs. 267 - 269
#9 – 19 odd,
27 – 39 odd,
59 – 67 EOO